Spherical waves oscillate in space and time with amplitudes that remain constant over any spherical surface centered on the source. The wave function of a spherical wave can be written as S(r, t) = sm (r)cos (kr - ωt + φ). Locations where two waves are perfectly in phase occur when the path difference is an integer multiple of the wavelength. Locations where two waves are perfectly out of phase occur when one path is an integer multiple of wavelengths and the other is a half integer multiple. When the frequency difference between two sound waves is small, a beat is heard; when the difference is large, two distinct tones are heard.

2. • Spherical waves are three dimensional waves.
• Spherical waves oscillate in space and time. However,
their amplitudes remain constant over any spherical
surface centered on the source.
• This allows us to write the wave function of a spherical
wave as
S(r, t) = sm (r)cos (kr - ωt + ϕ)

3. • Locations where two waves are perfectly in phase
• Determined to occur whenever the path difference is
an integer multiple of the wavelength
• Resultant waves can be determined using the formula
S(d, t)= 2sm(d)cos (kd -ωt + ϕ)

4. • Locations where two waves are perfectly out of phase
• Determined to occur when one path is an integer
multiple of wavelengths and the other is a half integer
multiple
• In other words, whenever
Δd= d2 – d1= (n+ 0.5) λ

5. • Two waves with slightly different frequencies have
variation of amplitude which results in a beat.
• When the frequency difference between two sound
waves is very large then we hear two distinct tones
rather than one that varies in intensity
• To determine the resultant wave the equation
Stotal(0, t)=2smcos(ϖt)cos(∆ωt)
can be used with the following quantities
and

6. • The following slide is a video using piano notes to
explain the concepts of consonance, dissonance, beats,
and interference
• Review: All musical notes have their own unique
frequencies.

7. • Question 1:
Two piano keys produce the frequencies of 262
Hz (C) and 330 Hz (G). What is the beat
frequency?

8. • Answer:
330 Hz- 262 Hz= 68 Hz

9. • Question 2:
Why don't we hear beats when different keys
on the piano are played at the same time?

10. • Answer:
In order to hear beats, two interfering sound
waves must have a difference in frequency of 7
Hz or less. No two keys on the piano produce
such a frequency.

11. • Question 3:
If a tuning fork with a frequency of 300 Hz is
played simultaneously with a note with a
frequency of 294 Hz (D). How many beats will
be heard over a period of 10 seconds?

12. • Answer:
300 Hz- 294 Hz= 6 Hz
In 10 seconds, there will be 60 beats.

13. Hawkes Et Al. Physics for Scientists and
Engineers: An Interactive Approach. Vol. 1.
Vancouver: U of British Columbia, n.d. Print.
Thank you for watching!