The document discusses the concepts of transverse and longitudinal waves, detailing their characteristics, equations, and behaviors when encountering boundaries. It covers topics such as wave speed, harmonic waves, standing waves, and acoustic waves, explaining phenomena like reflection, transmission, and resonance. Additionally, it highlights the significance of impedance matching in wave energy propagation across different media.

1. PHY
UNIT 2
NON-DISPERSIVE TRANSVERSE AND LONGITUDINAL WAVES
2.1 Vibrate the string up and down. The wave that developed due to this
vibration is called a transverse wave:
 Assume you are holding one side of a string and the other side is fixed and the
string is pulled fitted.
 Vibrate the string up and down. The wave that developed due to this vibration
is called a transverse wave.
 The wave in which the direction of disturbance is perpendicular to the direction
of travelling wave.
 The above diagram show that propagation takes place in the rope. The wave
travelled from one end to the other while the rope moved up and down.

2. Speed of a Wave on a String
 Velocity= distance travelled/Time
 In waves: Velocity can be found by dividing the wavelength(λ) by the
period(T):
v=λ/T We can take the inverse proportionality to period(T) and frequency(λ) and
apply it,
v=λ/T
v=λ1/T
v=λf
Speed of a Wave on a Vibrating String
 In musical instruments we can find wave in vibrating string like Keyboard and
guitar.
 In these cases, the frequency is what characterizes the pitch and therefore the
note.
 The rapidity of a wave on this kind of string is proportional to the square root
of the tension(T) in the string and inversely proportional to the square root
of the linear density of the string:v=√T/μ
2.2 The Wave Equation:
The wave equation for a plane wave traveling in the x- direction which is
Here, v is the phase velocity of the wave and y represents the variable which is
changing as the wave passes.

3.  This is the method of the wave equation which applies to a stretched string or a
plane electromagnetic (EM) wave.
 The mathematical description of a wave makes use of partial derivatives.
In 2 dimensions, the wave equation takes the form
Whichmight describe a wave on a stretched membrane.
Reflections
When transverse waves in strings meet one end they get reflected and when the
incident wave meets the reflected wave, interference happens.
2.3 Harmonic Waves,
Standing Waves and
Their Eigen
Frequencies:
A harmonic wave is a wave with a frequency () that is a +ve integer of
variousfrequency () of the unique wave, known as the fundamental frequency. The
unique wave is also called the 1st harmonic, the subsequent harmonics are known as
higher harmonics.

4.  In different two media there is reflection and transmission.
 Reflected wave are those who come back to the original position after reflection
from boundary

5. Assume thin rope attached to a thick rope with each rope held at opposite ends by
people. And suppose that a vibration is introduced by the person holding the end of
the thin rope then vibration travel from thin end to thick end.
Upon
reaching
the
boundary, the usually two behaviours will happen.
 The wave which come back to the initial position are known as reflected wave.
 The vibration which goes to thick end after vibrating to the rope known as
transmitted pulse.
 The reflected vibration will be found to be inverted. During the interaction
between the two mediums at the boundary, the Ist
particle of the more dense medium
defeats the smaller mass of the last particle of the less dense medium.
 This causes an upward displaced pulse to become a downward displaced pulse.

6.  The denser medium on the other hand was at rest prior to the interaction. The
Ist
particle of this medium receives an upward pull when the incident pulse touches
the boundary.
 Since, the denser medium was initially at rest, an upward pull causes an upward
displacement.
 Meanwhile the particles in this medium are initially at rest, any change in their
state of motion would be in the similar direction as the displacement of the particles
of the incident pulse.
 Some of the most exciting impedance matching occurs when energy comes in
the form of a wave. Meanwhile there is an impedance mismatch between the water
and the wall, the wave energy is unable to couple into the wall, and so it reflects

7. back
 Standing waves are always associated with resonance. Resonance can be
recognized by anintense increase in amplitude of the resultant vibrations.
 Any system in which standing waves can form has numerous natural frequencies.
The set of entirely possible standing waves are known as the harmonics of a system.
x is the distance the point traveled from the wave's source.
t is the time elapsed.

8. y0 is the amplitude of the oscillations.
c is the speed of the wave.
 Natural frequency isalso known as eigenfrequency. Eigen frequency is that
frequency at which a system tends to oscillate in the absence of any driving or
damping force.
 The motion pattern of a system oscillating at its natural frequency is called the
regular mode.
2.4
Logitudinal
Wave, Acoustics Waves:

9. Longitudinal Wave Formula
y (x, t) =y0cos[ω(t−xc)]
Where,
y = displacement of the point on the traveling sound wave
x =distance the point traveled from the wave’s source
t =time elapsed
y0 = amplitude of the oscillations
c = speed of the wave
ω =angular frequency of the wave
x/c = (t) (time wave takes to travel the distance x).
Frequency (f) of the wave is given by the formula:
f=ω2π

10.  Acoustic waves are a type of energy propagation over a medium by means
of adiabatic compression besides decompression. Significant quantities for relating
acoustic waves are acoustic pressure, particle velocity, particle
displacement and acoustic intensity.
 Acoustic waves travel with a typical acoustic velocity that depends on the
medium they're transitory through.
 Some instances of acoustic waves are audible sound from a speaker (waves
traveling through air at the speed of sound), ground movement from
an earthquake (waves traveling through the earth), or ultrasound used for medical
imaging (waves traveling through the body).

11. 2.6 Reflection and Transmission of waves at a boundary
 In different two media there is reflection and transmission.
 Reflected wave are those who come back to the original position after reflection
from boundary

12. Assume thin rope attached to a thick rope with each rope held at opposite ends by
people. And suppose that a vibration is introduced by the person holding the end of
the thin rope then vibration travel from thin end to thick end.
Upon
reaching the boundary, the usually two behaviours will happen.
 The wave which come back to the initial position are known as reflected wave.
 The vibration which goes to thick end after vibrating to the rope known as
transmitted pulse.

13.  The reflected vibration will be found to be inverted. During the interaction
between the two medium at the boundary, the Ist
particle of the more dense medium
defeats the smaller mass of the last particle of the less dense medium.
 This causes an upward displaced pulse to become a downward displaced pulse.
 The more dense medium on the other hand was at rest prior to the interaction.
The Ist
particle of this medium receives an upward pull when the incident pulse
touches the boundary.
 Since, the more dense medium was initially at rest, an upward pull cause an
upward displacement.
 Meanwhile the particles in this medium are initially at rest, any change in their
state of motion would be in the similar direction as the displacement of the particles
of the incident pulse.
 Some of the most exciting impedance matching occurs when energy comes in
the form of a wave. Meanwhile there is an impedance mismatch between the water
and the wall, the wave energy is unable to couple into the wall, and so it reflects
back.

14.  Standing waves are always associated with resonance. Resonance can be
recognized by a intense increase in amplitude of the resultant vibrations.
 Any system in which standing waves can form has numerous natural frequencies.
The set of entirely possible standing waves are known as the harmonics of a system.
x is the distance the point traveled from the wave's source.
t is the time elapsed.
y0 is the amplitude of the oscillations.
c is the speed of the wave.

15. 2.7 Impedance Matching
 Some of the most exciting impedance matching occurs when energy comes in
the form of a wave. Meanwhile there is an impedance mismatch between the water
and the wall, the wave energy is unable to couple into the wall, and so it reflects
back.