Aula 7b - ondas. Efeito Doppler e ondas emitidas por fontes supersônicas

1. Aula 7b -Ondas
• Ondas emitidas por fontes supersônicas
• Efeito Doppler Luminoso

2. <latexit sha1_base64="ZHNFm7aul4+awZu3cbU6m4LJoVo=">AAACCnicbVDLSsNAFJ3UV62vqEs3g0VwVRIRdFl047KCtYWmhMl02g6dR5iZFEPI1o2/4saFgrj1C9z5N07TLLT1wIXDOffO3HuimFFtPO/bqaysrq1vVDdrW9s7u3vu/sG9lonCpI0lk6obIU0YFaRtqGGkGyuCeMRIJ5pcz/zOlChNpbgzaUz6HI0EHVKMjJVCFwbFG5kigxxOwyzgkXzIAkNFCrXkeR66da/hFYDLxC9JHZRohe5XMJA44UQYzJDWPd+LTT9DylDMSF4LEk1ihCdoRHqWCsSJ7mfFFjk8scoADqWyJQws1N8TGeJapzyynRyZsV70ZuJ/Xi8xw8t+RkWcGCLw/KNhwqCRcBYLHFBFsGGpJQgraneFeIwUwsaGV7Mh+IsnL5POWcM/b/j+7Xm9eVXmUQVH4BicAh9cgCa4AS3QBhg8gmfwCt6cJ+fFeXc+5q0Vp5w5BH/gfP4Al8ebWA==</latexit>
vsom
<latexit sha1_base64="OoHnUMTfKHKutCSpgX5+xyUTuug=">AAACCHicbVDLSsNAFJ34rPUVdSnIYBFclUQKuiwK4rKCtYUmhMlk0g6dTMLMpBhCdm78FTcuFMStn+DOv3GaZqGtBy4czrl35t7jJ4xKZVnfxtLyyuraem2jvrm1vbNr7u3fyzgVmHRxzGLR95EkjHLSVVQx0k8EQZHPSM8fX0393oQISWN+p7KEuBEachpSjJSWPPPIKd/IBQkKOPFyJ/Ljh9xRlGfwuig8s2E1rRJwkdgVaYAKHc/8coIYpxHhCjMk5cC2EuXmSCiKGSnqTipJgvAYDclAU44iIt283KGAJ1oJYBgLXVzBUv09kaNIyizydWeE1EjOe1PxP2+QqvDCzSlPUkU4nn0UpgyqGE5DgQEVBCuWaYKwoHpXiEdIIKx0dHUdgj1/8iLpnTXtVtO2b1uN9mWVRw0cgmNwCmxwDtrgBnRAF2DwCJ7BK3gznowX4934mLUuGdXMAfgD4/MHoUGaOw==</latexit>
vF
<latexit sha1_base64="VHzpmnisfK3i/tDtRu5z2TTD20Q=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoBehqAePFVpbaELZbDft0s0m7E6EUvo3vHhQEK/+GG/+G7dtDtr6YODx3gwz88JUCoOu++0U1tY3NreK26Wd3b39g/Lh0aNJMs14iyUy0Z2QGi6F4i0UKHkn1ZzGoeTtcHQ789tPXBuRqCaOUx7EdKBEJBhFK/n+HZdICZJr0uyVK27VnYOsEi8nFcjR6JW//H7CspgrZJIa0/XcFIMJ1SiY5NOSnxmeUjaiA961VNGYm2Ayv3lKzqzSJ1GibSkkc/X3xITGxozj0HbGFIdm2ZuJ/3ndDKOrYCJUmiFXbLEoyiTBhMwCIH2hOUM5toQyLeythA2ppgxtTCUbgrf88ippX1S9WtXzHmqV+k2eRxFO4BTOwYNLqMM9NKAFDFJ4hld4czLnxXl3PhatBSefOYY/cD5/AAXrkNY=</latexit>
t = T
<latexit sha1_base64="+8FUkWl9JVjn0Yhmp09+PMq7Ym0=">AAAB9XicbVA9SwNBEJ3zM8avqKXNYhCswl0IaCMEtbCMkJhAcoS9zV6yZG/v3J0LhJDfYWOhILb+Fzv/jZvkCk18MPB4b4aZeUEihUHX/XbW1jc2t7ZzO/ndvf2Dw8LR8aOJU814g8Uy1q2AGi6F4g0UKHkr0ZxGgeTNYHg785sjro2IVR3HCfcj2lciFIyilfzOHZdICZJrUq53C0W35M5BVomXkSJkqHULX51ezNKIK2SSGtP23AT9CdUomOTTfCc1PKFsSPu8bamiETf+ZH70lJxbpUfCWNtSSObq74kJjYwZR4HtjCgOzLI3E//z2imGV/5EqCRFrthiUZhKgjGZJUB6QnOGcmwJZVrYWwkbUE0Z2pzyNgRv+eVV0iyXvErJ8x4qxepNlkcOTuEMLsCDS6jCPdSgAQye4Ble4c0ZOS/Ou/OxaF1zspkT+APn8wd6aZES</latexit>
t = 2T
<latexit sha1_base64="O0RPvHRw/IJZ09PwLbtl4k96bM0=">AAAB9XicbVA9SwNBEN2LXzF+RS1tFoNgFe40oI0Q1MIyQmICyRH2NnPJkr29c3cuEEJ+h42Fgtj6X+z8N26SKzT6YODx3gwz84JECoOu++XkVlbX1jfym4Wt7Z3dveL+wYOJU82hwWMZ61bADEihoIECJbQSDSwKJDSD4c3Mb45AGxGrOo4T8CPWVyIUnKGV/M4tSGQU6RU9r3eLJbfszkH/Ei8jJZKh1i1+dnoxTyNQyCUzpu25CfoTplFwCdNCJzWQMD5kfWhbqlgExp/Mj57SE6v0aBhrWwrpXP05MWGRMeMosJ0Rw4FZ9mbif147xfDSnwiVpAiKLxaFqaQY01kCtCc0cJRjSxjXwt5K+YBpxtHmVLAheMsv/yXNs7JXKXvefaVUvc7yyJMjckxOiUcuSJXckRppEE4eyRN5Ia/OyHl23pz3RWvOyWYOyS84H99775ET</latexit>
t = 3T
<latexit sha1_base64="Ztha2fo3iGVJ75b//X8HaS3m094=">AAAB8XicbVBNSwMxFHzrZ61fVY9egkX0VDZS0GPRi8cK1lbapWSz2TY0yS5JVihLf4UXDwri1X/jzX9j2u5BWwcCw8w88t6EqeDG+v63t7K6tr6xWdoqb+/s7u1XDg4fTJJpylo0EYnuhMQwwRVrWW4F66SaERkK1g5HN1O//cS04Ym6t+OUBZIMFI85JdZJjz3hohFBZ/1K1a/5M6BlggtShQLNfuWrFyU0k0xZKogxXeynNsiJtpwKNin3MsNSQkdkwLqOKiKZCfLZwhN06pQIxYl2T1k0U39P5EQaM5ahS0pih2bRm4r/ed3MxldBzlWaWabo/KM4E8gmaHo9irhm1IqxI4Rq7nZFdEg0odZ1VHYl4MWTl0n7oobrNYzv6tXGddFHCY7hBM4BwyU04Baa0AIKEp7hFd487b14797HPLriFTNH8Afe5w+QkZAR</latexit>
0
<latexit sha1_base64="Ztha2fo3iGVJ75b//X8HaS3m094=">AAAB8XicbVBNSwMxFHzrZ61fVY9egkX0VDZS0GPRi8cK1lbapWSz2TY0yS5JVihLf4UXDwri1X/jzX9j2u5BWwcCw8w88t6EqeDG+v63t7K6tr6xWdoqb+/s7u1XDg4fTJJpylo0EYnuhMQwwRVrWW4F66SaERkK1g5HN1O//cS04Ym6t+OUBZIMFI85JdZJjz3hohFBZ/1K1a/5M6BlggtShQLNfuWrFyU0k0xZKogxXeynNsiJtpwKNin3MsNSQkdkwLqOKiKZCfLZwhN06pQIxYl2T1k0U39P5EQaM5ahS0pih2bRm4r/ed3MxldBzlWaWabo/KM4E8gmaHo9irhm1IqxI4Rq7nZFdEg0odZ1VHYl4MWTl0n7oobrNYzv6tXGddFHCY7hBM4BwyU04Baa0AIKEp7hFd487b14797HPLriFTNH8Afe5w+QkZAR</latexit>
0
<latexit sha1_base64="Ztha2fo3iGVJ75b//X8HaS3m094=">AAAB8XicbVBNSwMxFHzrZ61fVY9egkX0VDZS0GPRi8cK1lbapWSz2TY0yS5JVihLf4UXDwri1X/jzX9j2u5BWwcCw8w88t6EqeDG+v63t7K6tr6xWdoqb+/s7u1XDg4fTJJpylo0EYnuhMQwwRVrWW4F66SaERkK1g5HN1O//cS04Ym6t+OUBZIMFI85JdZJjz3hohFBZ/1K1a/5M6BlggtShQLNfuWrFyU0k0xZKogxXeynNsiJtpwKNin3MsNSQkdkwLqOKiKZCfLZwhN06pQIxYl2T1k0U39P5EQaM5ahS0pih2bRm4r/ed3MxldBzlWaWabo/KM4E8gmaHo9irhm1IqxI4Rq7nZFdEg0odZ1VHYl4MWTl0n7oobrNYzv6tXGddFHCY7hBM4BwyU04Baa0AIKEp7hFd487b14797HPLriFTNH8Afe5w+QkZAR</latexit>
0
EFEITO DOPPLER:
LEMBRANDO DO QUE OCORRIA QUANTO A FONTE SE MOVIMENTAVA
Quando discutimos o efeito Doppler
devido ao movimento das fontes
observamos que a mudança de
frequência se dava porque a fonte
estava mudando de lugar e gerando
cada nova frente de onda num ponto
diferente no espaço. Assim as
frentes de ondas acabavam todas
excêntricas.

3. FONTE EM MOVIMENTO SUPERSÔNICO E OBSERVADOR EM REPOUSO
<latexit sha1_base64="OoHnUMTfKHKutCSpgX5+xyUTuug=">AAACCHicbVDLSsNAFJ34rPUVdSnIYBFclUQKuiwK4rKCtYUmhMlk0g6dTMLMpBhCdm78FTcuFMStn+DOv3GaZqGtBy4czrl35t7jJ4xKZVnfxtLyyuraem2jvrm1vbNr7u3fyzgVmHRxzGLR95EkjHLSVVQx0k8EQZHPSM8fX0393oQISWN+p7KEuBEachpSjJSWPPPIKd/IBQkKOPFyJ/Ljh9xRlGfwuig8s2E1rRJwkdgVaYAKHc/8coIYpxHhCjMk5cC2EuXmSCiKGSnqTipJgvAYDclAU44iIt283KGAJ1oJYBgLXVzBUv09kaNIyizydWeE1EjOe1PxP2+QqvDCzSlPUkU4nn0UpgyqGE5DgQEVBCuWaYKwoHpXiEdIIKx0dHUdgj1/8iLpnTXtVtO2b1uN9mWVRw0cgmNwCmxwDtrgBnRAF2DwCJ7BK3gznowX4934mLUuGdXMAfgD4/MHoUGaOw==</latexit>
vF
O movimento supersônico é o caso extremo deste
fenômeno. A fonte tem velocidade maior
que a velocidade de fase no gás e de fato vence
a frente de onda que ela própria gerou.
<latexit sha1_base64="ZHNFm7aul4+awZu3cbU6m4LJoVo=">AAACCnicbVDLSsNAFJ3UV62vqEs3g0VwVRIRdFl047KCtYWmhMl02g6dR5iZFEPI1o2/4saFgrj1C9z5N07TLLT1wIXDOffO3HuimFFtPO/bqaysrq1vVDdrW9s7u3vu/sG9lonCpI0lk6obIU0YFaRtqGGkGyuCeMRIJ5pcz/zOlChNpbgzaUz6HI0EHVKMjJVCFwbFG5kigxxOwyzgkXzIAkNFCrXkeR66da/hFYDLxC9JHZRohe5XMJA44UQYzJDWPd+LTT9DylDMSF4LEk1ihCdoRHqWCsSJ7mfFFjk8scoADqWyJQws1N8TGeJapzyynRyZsV70ZuJ/Xi8xw8t+RkWcGCLw/KNhwqCRcBYLHFBFsGGpJQgraneFeIwUwsaGV7Mh+IsnL5POWcM/b/j+7Xm9eVXmUQVH4BicAh9cgCa4AS3QBhg8gmfwCt6cJ+fFeXc+5q0Vp5w5BH/gfP4Al8ebWA==</latexit>
vsom

4. <latexit sha1_base64="OoHnUMTfKHKutCSpgX5+xyUTuug=">AAACCHicbVDLSsNAFJ34rPUVdSnIYBFclUQKuiwK4rKCtYUmhMlk0g6dTMLMpBhCdm78FTcuFMStn+DOv3GaZqGtBy4czrl35t7jJ4xKZVnfxtLyyuraem2jvrm1vbNr7u3fyzgVmHRxzGLR95EkjHLSVVQx0k8EQZHPSM8fX0393oQISWN+p7KEuBEachpSjJSWPPPIKd/IBQkKOPFyJ/Ljh9xRlGfwuig8s2E1rRJwkdgVaYAKHc/8coIYpxHhCjMk5cC2EuXmSCiKGSnqTipJgvAYDclAU44iIt283KGAJ1oJYBgLXVzBUv09kaNIyizydWeE1EjOe1PxP2+QqvDCzSlPUkU4nn0UpgyqGE5DgQEVBCuWaYKwoHpXiEdIIKx0dHUdgj1/8iLpnTXtVtO2b1uN9mWVRw0cgmNwCmxwDtrgBnRAF2DwCJ7BK3gznowX4934mLUuGdXMAfgD4/MHoUGaOw==</latexit>
vF
FONTE EM MOVIMENTO SUPERSÔNICO E OBSERVADOR EM REPOUSO
A fonte ganha da frente de onda!
<latexit sha1_base64="ZHNFm7aul4+awZu3cbU6m4LJoVo=">AAACCnicbVDLSsNAFJ3UV62vqEs3g0VwVRIRdFl047KCtYWmhMl02g6dR5iZFEPI1o2/4saFgrj1C9z5N07TLLT1wIXDOffO3HuimFFtPO/bqaysrq1vVDdrW9s7u3vu/sG9lonCpI0lk6obIU0YFaRtqGGkGyuCeMRIJ5pcz/zOlChNpbgzaUz6HI0EHVKMjJVCFwbFG5kigxxOwyzgkXzIAkNFCrXkeR66da/hFYDLxC9JHZRohe5XMJA44UQYzJDWPd+LTT9DylDMSF4LEk1ihCdoRHqWCsSJ7mfFFjk8scoADqWyJQws1N8TGeJapzyynRyZsV70ZuJ/Xi8xw8t+RkWcGCLw/KNhwqCRcBYLHFBFsGGpJQgraneFeIwUwsaGV7Mh+IsnL5POWcM/b/j+7Xm9eVXmUQVH4BicAh9cgCa4AS3QBhg8gmfwCt6cJ+fFeXc+5q0Vp5w5BH/gfP4Al8ebWA==</latexit>
vsom

5. <latexit sha1_base64="ZHNFm7aul4+awZu3cbU6m4LJoVo=">AAACCnicbVDLSsNAFJ3UV62vqEs3g0VwVRIRdFl047KCtYWmhMl02g6dR5iZFEPI1o2/4saFgrj1C9z5N07TLLT1wIXDOffO3HuimFFtPO/bqaysrq1vVDdrW9s7u3vu/sG9lonCpI0lk6obIU0YFaRtqGGkGyuCeMRIJ5pcz/zOlChNpbgzaUz6HI0EHVKMjJVCFwbFG5kigxxOwyzgkXzIAkNFCrXkeR66da/hFYDLxC9JHZRohe5XMJA44UQYzJDWPd+LTT9DylDMSF4LEk1ihCdoRHqWCsSJ7mfFFjk8scoADqWyJQws1N8TGeJapzyynRyZsV70ZuJ/Xi8xw8t+RkWcGCLw/KNhwqCRcBYLHFBFsGGpJQgraneFeIwUwsaGV7Mh+IsnL5POWcM/b/j+7Xm9eVXmUQVH4BicAh9cgCa4AS3QBhg8gmfwCt6cJ+fFeXc+5q0Vp5w5BH/gfP4Al8ebWA==</latexit>
vsom
<latexit sha1_base64="OoHnUMTfKHKutCSpgX5+xyUTuug=">AAACCHicbVDLSsNAFJ34rPUVdSnIYBFclUQKuiwK4rKCtYUmhMlk0g6dTMLMpBhCdm78FTcuFMStn+DOv3GaZqGtBy4czrl35t7jJ4xKZVnfxtLyyuraem2jvrm1vbNr7u3fyzgVmHRxzGLR95EkjHLSVVQx0k8EQZHPSM8fX0393oQISWN+p7KEuBEachpSjJSWPPPIKd/IBQkKOPFyJ/Ljh9xRlGfwuig8s2E1rRJwkdgVaYAKHc/8coIYpxHhCjMk5cC2EuXmSCiKGSnqTipJgvAYDclAU44iIt283KGAJ1oJYBgLXVzBUv09kaNIyizydWeE1EjOe1PxP2+QqvDCzSlPUkU4nn0UpgyqGE5DgQEVBCuWaYKwoHpXiEdIIKx0dHUdgj1/8iLpnTXtVtO2b1uN9mWVRw0cgmNwCmxwDtrgBnRAF2DwCJ7BK3gznowX4934mLUuGdXMAfgD4/MHoUGaOw==</latexit>
vF
<latexit sha1_base64="WtjzinxrlU4kzVpXGQRBeUYiwDI=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0ItQ1IPHCtYWmlA22027dLMJuxOhhP4NLx4UxKs/xpv/xm2bg7Y+GHi8N8PMvDCVwqDrfjulldW19Y3yZmVre2d3r7p/8GiSTDPeYolMdCekhkuheAsFSt5JNadxKHk7HN1M/fYT10Yk6gHHKQ9iOlAiEoyilXz/lkukBMkVcXvVmlt3ZyDLxCtIDQo0e9Uvv5+wLOYKmaTGdD03xSCnGgWTfFLxM8NTykZ0wLuWKhpzE+SzmyfkxCp9EiXalkIyU39P5DQ2ZhyHtjOmODSL3lT8z+tmGF0GuVBphlyx+aIokwQTMg2A9IXmDOXYEsq0sLcSNqSaMrQxVWwI3uLLy6R9VvfO6553f15rXBd5lOEIjuEUPLiABtxBE1rAIIVneIU3J3NenHfnY95acoqZQ/gD5/MHzyiQsg==</latexit>
t = 0
<latexit sha1_base64="VHzpmnisfK3i/tDtRu5z2TTD20Q=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoBehqAePFVpbaELZbDft0s0m7E6EUvo3vHhQEK/+GG/+G7dtDtr6YODx3gwz88JUCoOu++0U1tY3NreK26Wd3b39g/Lh0aNJMs14iyUy0Z2QGi6F4i0UKHkn1ZzGoeTtcHQ789tPXBuRqCaOUx7EdKBEJBhFK/n+HZdICZJr0uyVK27VnYOsEi8nFcjR6JW//H7CspgrZJIa0/XcFIMJ1SiY5NOSnxmeUjaiA961VNGYm2Ayv3lKzqzSJ1GibSkkc/X3xITGxozj0HbGFIdm2ZuJ/3ndDKOrYCJUmiFXbLEoyiTBhMwCIH2hOUM5toQyLeythA2ppgxtTCUbgrf88ippX1S9WtXzHmqV+k2eRxFO4BTOwYNLqMM9NKAFDFJ4hld4czLnxXl3PhatBSefOYY/cD5/AAXrkNY=</latexit>
t = T
<latexit sha1_base64="+8FUkWl9JVjn0Yhmp09+PMq7Ym0=">AAAB9XicbVA9SwNBEJ3zM8avqKXNYhCswl0IaCMEtbCMkJhAcoS9zV6yZG/v3J0LhJDfYWOhILb+Fzv/jZvkCk18MPB4b4aZeUEihUHX/XbW1jc2t7ZzO/ndvf2Dw8LR8aOJU814g8Uy1q2AGi6F4g0UKHkr0ZxGgeTNYHg785sjro2IVR3HCfcj2lciFIyilfzOHZdICZJrUq53C0W35M5BVomXkSJkqHULX51ezNKIK2SSGtP23AT9CdUomOTTfCc1PKFsSPu8bamiETf+ZH70lJxbpUfCWNtSSObq74kJjYwZR4HtjCgOzLI3E//z2imGV/5EqCRFrthiUZhKgjGZJUB6QnOGcmwJZVrYWwkbUE0Z2pzyNgRv+eVV0iyXvErJ8x4qxepNlkcOTuEMLsCDS6jCPdSgAQye4Ble4c0ZOS/Ou/OxaF1zspkT+APn8wd6aZES</latexit>
t = 2T
<latexit sha1_base64="O0RPvHRw/IJZ09PwLbtl4k96bM0=">AAAB9XicbVA9SwNBEN2LXzF+RS1tFoNgFe40oI0Q1MIyQmICyRH2NnPJkr29c3cuEEJ+h42Fgtj6X+z8N26SKzT6YODx3gwz84JECoOu++XkVlbX1jfym4Wt7Z3dveL+wYOJU82hwWMZ61bADEihoIECJbQSDSwKJDSD4c3Mb45AGxGrOo4T8CPWVyIUnKGV/M4tSGQU6RU9r3eLJbfszkH/Ei8jJZKh1i1+dnoxTyNQyCUzpu25CfoTplFwCdNCJzWQMD5kfWhbqlgExp/Mj57SE6v0aBhrWwrpXP05MWGRMeMosJ0Rw4FZ9mbif147xfDSnwiVpAiKLxaFqaQY01kCtCc0cJRjSxjXwt5K+YBpxtHmVLAheMsv/yXNs7JXKXvefaVUvc7yyJMjckxOiUcuSJXckRppEE4eyRN5Ia/OyHl23pz3RWvOyWYOyS84H99775ET</latexit>
t = 3T
<latexit sha1_base64="ZHNFm7aul4+awZu3cbU6m4LJoVo=">AAACCnicbVDLSsNAFJ3UV62vqEs3g0VwVRIRdFl047KCtYWmhMl02g6dR5iZFEPI1o2/4saFgrj1C9z5N07TLLT1wIXDOffO3HuimFFtPO/bqaysrq1vVDdrW9s7u3vu/sG9lonCpI0lk6obIU0YFaRtqGGkGyuCeMRIJ5pcz/zOlChNpbgzaUz6HI0EHVKMjJVCFwbFG5kigxxOwyzgkXzIAkNFCrXkeR66da/hFYDLxC9JHZRohe5XMJA44UQYzJDWPd+LTT9DylDMSF4LEk1ihCdoRHqWCsSJ7mfFFjk8scoADqWyJQws1N8TGeJapzyynRyZsV70ZuJ/Xi8xw8t+RkWcGCLw/KNhwqCRcBYLHFBFsGGpJQgraneFeIwUwsaGV7Mh+IsnL5POWcM/b/j+7Xm9eVXmUQVH4BicAh9cgCa4AS3QBhg8gmfwCt6cJ+fFeXc+5q0Vp5w5BH/gfP4Al8ebWA==</latexit>
vsom
<latexit sha1_base64="ZHNFm7aul4+awZu3cbU6m4LJoVo=">AAACCnicbVDLSsNAFJ3UV62vqEs3g0VwVRIRdFl047KCtYWmhMl02g6dR5iZFEPI1o2/4saFgrj1C9z5N07TLLT1wIXDOffO3HuimFFtPO/bqaysrq1vVDdrW9s7u3vu/sG9lonCpI0lk6obIU0YFaRtqGGkGyuCeMRIJ5pcz/zOlChNpbgzaUz6HI0EHVKMjJVCFwbFG5kigxxOwyzgkXzIAkNFCrXkeR66da/hFYDLxC9JHZRohe5XMJA44UQYzJDWPd+LTT9DylDMSF4LEk1ihCdoRHqWCsSJ7mfFFjk8scoADqWyJQws1N8TGeJapzyynRyZsV70ZuJ/Xi8xw8t+RkWcGCLw/KNhwqCRcBYLHFBFsGGpJQgraneFeIwUwsaGV7Mh+IsnL5POWcM/b/j+7Xm9eVXmUQVH4BicAh9cgCa4AS3QBhg8gmfwCt6cJ+fFeXc+5q0Vp5w5BH/gfP4Al8ebWA==</latexit>
vsom
Cone de Mach
FONTE EM MOVIMENTO SUPERSÔNICO E OBSERVADOR EM REPOUSO

6. <latexit sha1_base64="ZHNFm7aul4+awZu3cbU6m4LJoVo=">AAACCnicbVDLSsNAFJ3UV62vqEs3g0VwVRIRdFl047KCtYWmhMl02g6dR5iZFEPI1o2/4saFgrj1C9z5N07TLLT1wIXDOffO3HuimFFtPO/bqaysrq1vVDdrW9s7u3vu/sG9lonCpI0lk6obIU0YFaRtqGGkGyuCeMRIJ5pcz/zOlChNpbgzaUz6HI0EHVKMjJVCFwbFG5kigxxOwyzgkXzIAkNFCrXkeR66da/hFYDLxC9JHZRohe5XMJA44UQYzJDWPd+LTT9DylDMSF4LEk1ihCdoRHqWCsSJ7mfFFjk8scoADqWyJQws1N8TGeJapzyynRyZsV70ZuJ/Xi8xw8t+RkWcGCLw/KNhwqCRcBYLHFBFsGGpJQgraneFeIwUwsaGV7Mh+IsnL5POWcM/b/j+7Xm9eVXmUQVH4BicAh9cgCa4AS3QBhg8gmfwCt6cJ+fFeXc+5q0Vp5w5BH/gfP4Al8ebWA==</latexit>
vsom
<latexit sha1_base64="OoHnUMTfKHKutCSpgX5+xyUTuug=">AAACCHicbVDLSsNAFJ34rPUVdSnIYBFclUQKuiwK4rKCtYUmhMlk0g6dTMLMpBhCdm78FTcuFMStn+DOv3GaZqGtBy4czrl35t7jJ4xKZVnfxtLyyuraem2jvrm1vbNr7u3fyzgVmHRxzGLR95EkjHLSVVQx0k8EQZHPSM8fX0393oQISWN+p7KEuBEachpSjJSWPPPIKd/IBQkKOPFyJ/Ljh9xRlGfwuig8s2E1rRJwkdgVaYAKHc/8coIYpxHhCjMk5cC2EuXmSCiKGSnqTipJgvAYDclAU44iIt283KGAJ1oJYBgLXVzBUv09kaNIyizydWeE1EjOe1PxP2+QqvDCzSlPUkU4nn0UpgyqGE5DgQEVBCuWaYKwoHpXiEdIIKx0dHUdgj1/8iLpnTXtVtO2b1uN9mWVRw0cgmNwCmxwDtrgBnRAF2DwCJ7BK3gznowX4934mLUuGdXMAfgD4/MHoUGaOw==</latexit>
vF
Cone de Mach
<latexit sha1_base64="J/s2/RQEDyBbtvZRKL0voTiJXMs=">AAACAHicbVDLSsNAFJ3UV62vquDGzWARXJVECrosunFZobVCE8JkOmmHziPMTIohZuGvuHGhIG79DHf+jdPHQlsPXDiccy/33hMljGrjut9OaWV1bX2jvFnZ2t7Z3avuH9xpmSpMOlgyqe4jpAmjgnQMNYzcJ4ogHjHSjUbXE787JkpTKdomS0jA0UDQmGJkrBRWj8Zh7vNIPuS+oSKDWvKigO2wWnPr7hRwmXhzUgNztMLql9+XOOVEGMyQ1j3PTUyQI2UoZqSo+KkmCcIjNCA9SwXiRAf59P4CnlqlD2OpbAkDp+rviRxxrTMe2U6OzFAvehPxP6+XmvgyyKlIUkMEni2KUwaNhJMwYJ8qgg3LLEFYUXsrxEOkEDY2sooNwVt8eZl0z+teo+55t41a82qeRxkcgxNwBjxwAZrgBrRAB2DwCJ7BK3hznpwX5935mLWWnPnMIfgD5/MHG8qWmg==</latexit>
vsomT
<latexit sha1_base64="SjkEQJa5K9d98WNNc6yQKXBRGtI=">AAAB/nicbVDLSsNAFJ3UV62v+Ni5GSyCq5JIQZdFQVxWaG2hDWEynbRDJzNhZlKMIfgrblwoiFu/w51/47TNQlsPXDiccy/33hPEjCrtON9WaWV1bX2jvFnZ2t7Z3bP3D+6VSCQmbSyYkN0AKcIoJ21NNSPdWBIUBYx0gvH11O9MiFRU8JZOY+JFaMhpSDHSRvLto4mf9aNAPGR9TXkKb/Ictny76tScGeAycQtSBQWavv3VHwicRIRrzJBSPdeJtZchqSlmJK/0E0VihMdoSHqGchQR5WWz63N4apQBDIU0xTWcqb8nMhQplUaB6YyQHqlFbyr+5/USHV56GeVxognH80VhwqAWcBoFHFBJsGapIQhLam6FeIQkwtoEVjEhuIsvL5POec2t11z3rl5tXBV5lMExOAFnwAUXoAFuQRO0AQaP4Bm8gjfryXqx3q2PeWvJKmYOwR9Ynz8s9pV9</latexit>
vFT
O ângulo de Mach é dado por
O número de Mach é definido
como a razão
<latexit sha1_base64="tn9WRz//X0O3rp/T53OPso/Rp4M=">AAACK3icbZDLSsNAFIYn9V5vVZduBovgqiRS0I3gBcRlBWuFJoTJdGIH5xJmTsQS+jxufBU3LhTFre/hpO1Cqz8M/HznHM6cP8kEt+D7715lZnZufmFxqbq8srq2XtvYvLY6N5S1qRba3CTEMsEVawMHwW4yw4hMBOskd2dlvXPPjOVaXcEgY5Ekt4qnnBJwKK6dhJYrHEKfAYklPsJhaggt7uMilIl+KELgalBYLYdO0/i8hHGt7jf8kfBfE0xMHU3UimsvYU/TXDIFVBBru4GfQVQQA5wKNqyGuWUZoXfklnWdVUQyGxWjU4d415EeTrVxTwEe0Z8TBZHWDmTiOiWBvp2ulfC/WjeH9DAquMpyYIqOF6W5wKBxmRvuccMoiIEzhBru/oppn7iswKVbdSEE0yf/NZ39RtBsBMFls358OsljEW2jHbSHAnSAjtEFaqE2ougRPaNX9OY9eS/eh/c5bq14k5kt9Eve1zeUXKpK</latexit>
sin ✓m =
vsom
vF
<latexit sha1_base64="bCu8ynkIwjDlNMfJbAaEGhzgGGw=">AAAB8HicbVA9SwNBEN2LXzF+RS1tFoNgFe4koGXQxjKCMYHkCHubuWTJ7t65OyeEkD9hY6Egtv4cO/+Nm+QKTXww8Hhvhpl5USqFRd//9gpr6xubW8Xt0s7u3v5B+fDowSaZ4dDkiUxMO2IWpNDQRIES2qkBpiIJrWh0M/NbT2CsSPQ9jlMIFRtoEQvO0EntLg4BWU/1yhW/6s9BV0mQkwrJ0eiVv7r9hGcKNHLJrO0EforhhBkUXMK01M0spIyP2AA6jmqmwIaT+b1TeuaUPo0T40ojnau/JyZMWTtWketUDId22ZuJ/3mdDOOrcCJ0miFovlgUZ5JiQmfP074wwFGOHWHcCHcr5UNmGEcXUcmFECy/vEpaF9WgVg2Cu1qlfp3nUSQn5JSck4Bckjq5JQ3SJJxI8kxeyZv36L14797HorXg5TPH5A+8zx/BRJBB</latexit>
✓m
.
<latexit sha1_base64="/QLsMUxHkj5wVR2oENVzFvt2zMk=">AAACIHicbZDLSgMxFIYz9VbrbdSlm2ARXJUZEXUjFAVxI1SwttAZSibNtKG5DEmmWIa+iRtfxY0LBdGdPo1pOwtt/SHw851zODl/lDCqjed9OYWFxaXlleJqaW19Y3PL3d651zJVmNSxZFI1I6QJo4LUDTWMNBNFEI8YaUT9y3G9MSBKUynuzDAhIUddQWOKkbGo7Z7AG3gOg1ghnA3aWcAj+ZAFhophdjWymoVa8jFuu2Wv4k0E542fmzLIVWu7n0FH4pQTYTBDWrd8LzFhhpShmJFRKUg1SRDuoy5pWSsQJzrMJveN4IElHRhLZZ8wcEJ/T2SIaz3kke3kyPT0bG0M/6u1UhOfhRkVSWqIwNNFccqgkXAcFuxQRbBhQ2sQVtT+FeIeslkZG2nJhuDPnjxvGkcV/7ji+7fH5epFnkcR7IF9cAh8cAqq4BrUQB1g8AiewSt4c56cF+fd+Zi2Fpx8Zhf8kfP9A+6upUU=</latexit>
M =
vF
vsom
FONTE EM MOVIMENTO SUPERSÔNICO E OBSERVADOR EM REPOUSO

7. <latexit sha1_base64="OoHnUMTfKHKutCSpgX5+xyUTuug=">AAACCHicbVDLSsNAFJ34rPUVdSnIYBFclUQKuiwK4rKCtYUmhMlk0g6dTMLMpBhCdm78FTcuFMStn+DOv3GaZqGtBy4czrl35t7jJ4xKZVnfxtLyyuraem2jvrm1vbNr7u3fyzgVmHRxzGLR95EkjHLSVVQx0k8EQZHPSM8fX0393oQISWN+p7KEuBEachpSjJSWPPPIKd/IBQkKOPFyJ/Ljh9xRlGfwuig8s2E1rRJwkdgVaYAKHc/8coIYpxHhCjMk5cC2EuXmSCiKGSnqTipJgvAYDclAU44iIt283KGAJ1oJYBgLXVzBUv09kaNIyizydWeE1EjOe1PxP2+QqvDCzSlPUkU4nn0UpgyqGE5DgQEVBCuWaYKwoHpXiEdIIKx0dHUdgj1/8iLpnTXtVtO2b1uN9mWVRw0cgmNwCmxwDtrgBnRAF2DwCJ7BK3gznowX4934mLUuGdXMAfgD4/MHoUGaOw==</latexit>
vF
FONTE EM MOVIMENTO SUPERSÔNICO E OBSERVADOR EM REPOUSO
O que um observador ouve?
fora do cone não ouve nada

8. <latexit sha1_base64="OoHnUMTfKHKutCSpgX5+xyUTuug=">AAACCHicbVDLSsNAFJ34rPUVdSnIYBFclUQKuiwK4rKCtYUmhMlk0g6dTMLMpBhCdm78FTcuFMStn+DOv3GaZqGtBy4czrl35t7jJ4xKZVnfxtLyyuraem2jvrm1vbNr7u3fyzgVmHRxzGLR95EkjHLSVVQx0k8EQZHPSM8fX0393oQISWN+p7KEuBEachpSjJSWPPPIKd/IBQkKOPFyJ/Ljh9xRlGfwuig8s2E1rRJwkdgVaYAKHc/8coIYpxHhCjMk5cC2EuXmSCiKGSnqTipJgvAYDclAU44iIt283KGAJ1oJYBgLXVzBUv09kaNIyizydWeE1EjOe1PxP2+QqvDCzSlPUkU4nn0UpgyqGE5DgQEVBCuWaYKwoHpXiEdIIKx0dHUdgj1/8iLpnTXtVtO2b1uN9mWVRw0cgmNwCmxwDtrgBnRAF2DwCJ7BK3gznowX4934mLUuGdXMAfgD4/MHoUGaOw==</latexit>
vF
FONTE EM MOVIMENTO SUPERSÔNICO E OBSERVADOR EM REPOUSO
O que um observador ouve?
Dentro do cone ele ouve uma
mistura de sons.
1) Som com frequência
emitido anteriormente e
cujas frentes de onda o atingem
(pela esquerda na figura) em ordem
inversa que foram emitidas.
2) Som com frequência
emitido posteriormente pela
fonte e cujas frentes de onda o
atingem (pela direita na figura)
na ordem correta que foram emitidas
<latexit sha1_base64="DxiwrYL99sraCKqtuFcum19JR+o=">AAACTnicbZHNSxtBGMZn41eM1UZ79PJikOrBsCtCvQhiofSYQmOEbAizk9lkcD6WmXfFsO4/6EXorf9FLx4siJ18HGr0hYGH3/O+vDPPJJkUDsPwd1BZWl5ZXauu1zY+bG59rG/vXDqTW8bbzEhjrxLquBSat1Gg5FeZ5VQlkneS668Tv3PDrRNG/8RxxnuKDrVIBaPoUb8+SD+fQQqx5CkeQJxaygq46RexSsxtEaPQY3BGlSUAlMXd+9bRIv7m4R2UEFsxHOFhv94Im+G04K2I5qJB5tXq13/FA8NyxTUySZ3rRmGGvYJaFEzyshbnjmeUXdMh73qpqeKuV0zTKGHfkwGkxvqjEab0/4mCKufGKvGdiuLILXoT+J7XzTE97RVCZzlyzWaL0lwCGphECwNhOUM59oIyK/xdgY2oTxT9B9R8CNHik9+KznEzOmlG0Y+TxvnFPI8q2SV75IBE5As5J99Ji7QJI/fkD3kif4OH4DF4Dl5mrZVgPvOJvKpK9R+3DLSl</latexit>
f0
= f
✓
vsom
|vsom vF|
◆
<latexit sha1_base64="vHRADjV6+NPGVsggsU3mr63dDmQ=">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</latexit>
f0
= f
✓
vsom
vsom + vF
◆

9. <latexit sha1_base64="OoHnUMTfKHKutCSpgX5+xyUTuug=">AAACCHicbVDLSsNAFJ34rPUVdSnIYBFclUQKuiwK4rKCtYUmhMlk0g6dTMLMpBhCdm78FTcuFMStn+DOv3GaZqGtBy4czrl35t7jJ4xKZVnfxtLyyuraem2jvrm1vbNr7u3fyzgVmHRxzGLR95EkjHLSVVQx0k8EQZHPSM8fX0393oQISWN+p7KEuBEachpSjJSWPPPIKd/IBQkKOPFyJ/Ljh9xRlGfwuig8s2E1rRJwkdgVaYAKHc/8coIYpxHhCjMk5cC2EuXmSCiKGSnqTipJgvAYDclAU44iIt283KGAJ1oJYBgLXVzBUv09kaNIyizydWeE1EjOe1PxP2+QqvDCzSlPUkU4nn0UpgyqGE5DgQEVBCuWaYKwoHpXiEdIIKx0dHUdgj1/8iLpnTXtVtO2b1uN9mWVRw0cgmNwCmxwDtrgBnRAF2DwCJ7BK3gznowX4934mLUuGdXMAfgD4/MHoUGaOw==</latexit>
vF
FONTE EM MOVIMENTO SUPERSÔNICO E OBSERVADOR EM REPOUSO
Quebra da barreira do som
Não existe um cone de Mach
mas um plano.
Quando a fonte passa pelo observador
ele ouve um ruído alto (estouro) devido
a superposição de todas as frentes
de ondas que se propagam (na
figura) para direita.
Depois ele percebe o som emitido
pela fonte (que se afasta) e que chega
nele se propagando para a esquerda
(na figura) e cuja frequência é a
metade da frequência realmente
emitida pela fonte.

10. Efeito Doppler da Luz
Não há meio de propagação. As ondas eletromagnéticas
se propagam no vácuo. Importa somente a velocidade
relativa entre fonte e observador
<latexit sha1_base64="94pNZEITFn/llWrV3hLk/Fnd0PY=">AAACE3icbZDNSgMxFIUz9a/Wv1GXboJFVMQ6kYKCCEU3LitYW+iUkkkzbWgmMyaZQhn6DG58FTcuFMStG3e+jZl2Ftp6IPBx7r3c3ONFnCntON9Wbm5+YXEpv1xYWV1b37A3t+5VGEtCayTkoWx4WFHOBK1ppjltRJLiwOO07vWv03p9QKViobjTw4i2AtwVzGcEa2O17UN//xL60L2ArnqQ2vUlJglE8AgOTggcpXg8wbZddErOWHAWUAZFkKnatr/cTkjigApNOFaqiZxItxIsNSOcjgpurGiESR93adOgwAFVrWR80gjuGacD/VCaJzQcu78nEhwoNQw80xlg3VPTtdT8r9aMtX/eSpiIYk0FmSzyYw51CNN8YIdJSjQfGsBEMvNXSHrYpKJNigUTApo+eRbqpyVULiF0Wy5WrrI88mAH7IIDgMAZqIAbUAU1QMAjeAav4M16sl6sd+tj0pqzsplt8EfW5w88Q5p9</latexit>
f0
= f
s
1 + v/c
1 v/c
A expressão Doppler Relativística para a frequência observada é
onde “v" é a velocidade relativa entre fonte e observador e seu sinal indica se estão
se aproximando (+) ou se afastando (-) e “c" é a velocidade da luz.
<latexit sha1_base64="bBl4i2oJ8KP1Oode6I/v2G4pSv4=">AAACBXicbVDLSgMxFM3UV62vUZfdBIvgqs6USluKUHTjsoK1hc5QMmmmDU0yQ5IRytCFG3/FjQsFces/uPNvTB8LbT1wL4dz7iW5J4gZVdpxvq3M2vrG5lZ2O7ezu7d/YB8e3asokZi0cMQi2QmQIowK0tJUM9KJJUE8YKQdjK6nfvuBSEUjcafHMfE5GggaUoy0kXp2HsNLWKrVoFev1EpeHZYvqqZzeA5Vzy44RWcGuErcBSmABZo9+8vrRzjhRGjMkFJd14m1nyKpKWZkkvMSRWKER2hAuoYKxIny09kRE3hqlD4MI2lKaDhTf2+kiCs15oGZ5EgP1bI3Ff/zuokOq35KRZxoIvD8oTBhUEdwmgjsU0mwZmNDEJbU/BXiIZIIa5NbzoTgLp+8Stqlolsuuu5tudC4WuSRBXlwAs6ACyqgAW5AE7QABo/gGbyCN+vJerHerY/5aMZa7ByDP7A+fwDIhZQt</latexit>
c = 299 792 458 m/s

11. Efeito Doppler da Luz
Desvio para o vermelho ( REDSHIFT )
Uma fonte de luz vai parecer mais avermelhada* quando for
for vista se afastando do observador
<latexit sha1_base64="7WoEwgk+3mN0flfHUZtQdxHTihU=">AAAB73icbVBNS8NAEJ34WetX1aOXxSJ6KokU9OCh6MVjBWsKbSib7aRdutmE3Y1QQn+EFw8K4tW/481/47bNQVsfDDzem2FmXpgKro3rfjsrq2vrG5ulrfL2zu7efuXg8FEnmWLYYolIVDukGgWX2DLcCGynCmkcCvTD0e3U959QaZ7IBzNOMYjpQPKIM2qs5Edn5JpEpFepujV3BrJMvIJUoUCzV/nq9hOWxSgNE1TrjuemJsipMpwJnJS7mcaUshEdYMdSSWPUQT47d0JOrdInUaJsSUNm6u+JnMZaj+PQdsbUDPWiNxX/8zqZia6CnMs0MyjZfFGUCWISMv2d9LlCZsTYEsoUt7cSNqSKMmMTKtsQvMWXl4l/UfPqNc+7r1cbN0UeJTiGEzgHDy6hAXfQhBYwGMEzvMKbkzovzrvzMW9dcYqZI/gD5/MHBvaOiA==</latexit>
f0
< f
* O efeito Doppler relativístico vale para todo o espectro de ondas-eletromagnéticas
não só para a porção visível ao olho humano. Assim “avermelhar" e “azular" neste contexto
devem ser considerados metaforicamente no sentido de abaixar ou aumentar a frequência.

12. Efeito Doppler da Luz
Desvio para o azul ( BLUESHIFT )
Uma fonte de luz vai parecer mais azulada* quando for
for vista se aproximando do observador
* O efeito Doppler relativístico vale para todo o espectro de ondas-eletromagnéticas
não só para a porção visível ao olho humano. Assim “avermelhar" e “azular" neste contexto
devem ser considerados metaforicamente no sentido de abaixar ou aumentar a frequência.
<latexit sha1_base64="vS0HrCHaQFArtjWxTTQu3ErfbnE=">AAAB73icbVBNS8NAEJ34WetX1aOXxSJ6KokU9CRFLx4rWFNoQ9lsJ+3SzSbsboQS+iO8eFAQr/4db/4bt20O2vpg4PHeDDPzwlRwbVz321lZXVvf2Cxtlbd3dvf2KweHjzrJFMMWS0Si2iHVKLjEluFGYDtVSONQoB+Obqe+/4RK80Q+mHGKQUwHkkecUWMlPzoj1yQivUrVrbkzkGXiFaQKBZq9yle3n7AsRmmYoFp3PDc1QU6V4UzgpNzNNKaUjegAO5ZKGqMO8tm5E3JqlT6JEmVLGjJTf0/kNNZ6HIe2M6ZmqBe9qfif18lMdBXkXKaZQcnmi6JMEJOQ6e+kzxUyI8aWUKa4vZWwIVWUGZtQ2YbgLb68TPyLmleved59vdq4KfIowTGcwDl4cAkNuIMmtIDBCJ7hFd6c1Hlx3p2PeeuKU8wcwR84nz8KBo6K</latexit>
f0
> f

13. Efeito Doppler da Luz
Aproximação para velocidades relativas pequenas em
relação à velocidade da Luz
<latexit sha1_base64="94pNZEITFn/llWrV3hLk/Fnd0PY=">AAACE3icbZDNSgMxFIUz9a/Wv1GXboJFVMQ6kYKCCEU3LitYW+iUkkkzbWgmMyaZQhn6DG58FTcuFMStG3e+jZl2Ftp6IPBx7r3c3ONFnCntON9Wbm5+YXEpv1xYWV1b37A3t+5VGEtCayTkoWx4WFHOBK1ppjltRJLiwOO07vWv03p9QKViobjTw4i2AtwVzGcEa2O17UN//xL60L2ArnqQ2vUlJglE8AgOTggcpXg8wbZddErOWHAWUAZFkKnatr/cTkjigApNOFaqiZxItxIsNSOcjgpurGiESR93adOgwAFVrWR80gjuGacD/VCaJzQcu78nEhwoNQw80xlg3VPTtdT8r9aMtX/eSpiIYk0FmSzyYw51CNN8YIdJSjQfGsBEMvNXSHrYpKJNigUTApo+eRbqpyVULiF0Wy5WrrI88mAH7IIDgMAZqIAbUAU1QMAjeAav4M16sl6sd+tj0pqzsplt8EfW5w88Q5p9</latexit>
f0
= f
s
1 + v/c
1 v/c
Se então a expressão
<latexit sha1_base64="/TzzH7hQKdcBYC2rHqz9iEgNAjA=">AAAB8HicbVA9SwNBEJ2LXzF+nVraLAbBKt5KQIsUQRvLCMYEkiPsbfaSJXt75+5eIBz5EzYWCmLrz7Hz37hJrtDEBwOP92aYmRckgmvjed9OYW19Y3OruF3a2d3bP3APjx51nCrKmjQWsWoHRDPBJWsabgRrJ4qRKBCsFYxuZ35rzJTmsXwwk4T5ERlIHnJKjJXa4wuKajWEe27Zq3hzoFWCc1KGHI2e+9XtxzSNmDRUEK072EuMnxFlOBVsWuqmmiWEjsiAdSyVJGLaz+b3TtGZVfoojJUtadBc/T2RkUjrSRTYzoiYoV72ZuJ/Xic14bWfcZmkhkm6WBSmApkYzZ5Hfa4YNWJiCaGK21sRHRJFqLERlWwIePnlVdK6rOBqBeP7arl+k+dRhBM4hXPAcAV1uIMGNIGCgGd4hTfnyXlx3p2PRWvByWeO4Q+czx/D3470</latexit>
v/c << 1
usando que e para
fica
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p
1 + x ' 1 + x/2
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x << 1
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f0
' f(1 + v/c)
O que é semelhante ao caso do Doppler no som para o caso que as velocidades
de fonte e de observador são muito menores que a velocidade do som.
1
p
1 x
' 1 + x/2