## Sound Waves Synopsis

**General Equations Of Wave Motion:**- In the positive y = f(vt- x); and in the negative r-direction, y = f(vt + x).
- Here v = velocity of wave propagation, x = position, and t = time at which the function y (displacement) is measured.
- In the sinusoidal form, y = Asin(ωt ± kx), where A = amplitude (maximum value of the periodic function), ω = angular frequency = \(\frac{2 \pi}{T}=2 \pi f\), T = time period (in s), and f = frequency

**Particle Velocity And Slope:**The particle velocity is given by

\(v_{\text {particle }}=\frac{d y}{d t}=A \omega \cos (\omega t \pm k x)=v_{\max } \cos (\omega t \pm k x)\)

The slope of a waveform is \(\frac{d y}{d x}= \pm A k \cos (\omega t \pm k x)\)

Thus, \(v_{\text {particle }}=v_{\text {wave }} \cdot \mid \text { slope } \mid\)- The resultant amplitude A with a phase difference Φ is given by

\(A^2=A_1^2+A_2^2+2 A_1 A_2 \cos \phi\)- For maxima, \(\phi=\left(\frac{2 \pi}{\lambda}\right) x= \pm 2 n \pi\)
- For minima, \(\phi=\left(\frac{2 \pi}{\lambda}\right) x= \pm(2 n+1) \pi\)

- Standing waves are produced by the superposition of identical waves traveling in opposite directions.

y = A sin(ωt – kx) + A sin(ωt + kx)

= 2A sin ωt cos kx

= (2A cos kx) sin ωf.

In a standing wave, all particles execute an SHM about their mean position with die same frequency but with the amplitude A(x) = 2A cos kx, which is position-dependent **Transverse Vibrations Of A String:**Velocity of wave = \(v=\sqrt{\frac{F}{\mu}}\) = where F = tension in string (in N) and

μ = mass per unit length (in kg m^{-1}).**Modes Of Vibrations In A Stretched String:**- Fundamental mode = first harmonic: \(f_1=\frac{1}{2 l} \sqrt{\frac{F}{\mu}}\)
- First overtone = second harmonic: \(f_2=\frac{2}{2 l} \sqrt{\frac{F}{\mu}}=2 f_1\)
- Second overtone = third harmonic: \(f_3=\frac{3}{2 l} \sqrt{\frac{F}{\mu}}=3 f_1\)
- (p-l)th overtone = pth harmonic: \(f_{\mathrm{P}}=\frac{p}{2 l} \sqrt{\frac{F}{\mu}}=p f_1\)

**Speed Of Sound Waves In A Gaseous Medium:**

\(\hat{v}=\sqrt{\frac{\gamma p}{\rho}}, \text { where } \gamma=\frac{c_p}{c_v}\), where \(\gamma=\frac{c_p}{c_v}\), p = pressure and p = density = \(\frac{M}{V}\)

Hence, \(v=\sqrt{\frac{\gamma p V}{M}}=\sqrt{\frac{\gamma R T}{M_0}}\), where M_{0}= molar mass.**Sound Waves Are Pressure Wives:**The excess pressure as a function

of x and t is

p = p_{0}cos (ωt- kx), where p_{0}= pressure amplitude.

Instanding waves, die pressure nodes and displacement antinodes are coincident.**The Vibration Of An Air Column (organ pipe):****Close Organ Pipe:**In a closed pipe, the open end is the pressure node as well as the displacement antinode.

**The Following Are The Modes Of Vibrations In A Closed Pipe:**

**Fundamental Or First Harmonic:**f_{1}= \(\frac{v}{4l}\)**First Overtone Or Third Harmonic:**\(f_3=3\left(\frac{v}{4 l}\right)=3 f_1\)**Second Overtone Or Fifth Harmonic:**\(f_5=5\left(\frac{v}{4 l}\right)=5 f_1\)

**Open Organ Pipe:****The Modes Of Vibrations In An Open Pipe Are As Follows:****Fundamental Or First Harmonic:**\(f_1=\frac{v}{2 l}\)**First Overtone Or Second Harmonic:**\(f_2=2\left(\frac{v}{2 l}\right)=2 f_1\)**Second Overtone Or Third Harmonic:**\(f_3=3\left(\frac{v}{2 l}\right)=3 f_1\)

- Note that, in a closed pipe only odd harmonics are present, while an open pipe contains all the harmonics. The richness of overtones in an open pipe makes the note melodious.
**Doppler Effect:**The apparent change in frequency due to the relative motion between the source of waves and the receiver is called the Doppler effect. A decrease in separation leads to an apparent increase in frequency. The general equation for the apparent frequency is

\(f^{\prime}=\left(\frac{v \pm v_0}{v \pm v_s}\right) f\)- where v = velocity of the sound wave, v
_{0}= velocity of the observer, v_{s}= velocity of the source,f = true frequency, and f’ = apparent frequency. **Beats:**Beats are the rhythmic variation of loudness at a point due to the superposition of waves having a small difference in their frequencies. This may be regarded as an interference in time (with the path difference fixed).- Beat frequency = difference in frequencies.
**The Intensity Of Sound Waves:**It is the amount of energy passing through a unit area per unit of time perpendicular to the area element. Thus, intensity = \(I=\frac{\Delta U}{\Delta A \Delta t}\) (**SI Unit:**W m^{-2}), and it is proportional- to the square of pressure amplitude.
**Loudness:**It is what we perceive as the volume of a sound.- The loudness level (β) is defined by the relation
- \(\beta=\log \frac{I}{I_0} B=10 \log \frac{I}{I_0} \mathrm{~dB}\)
- The minimum intensity (l
_{0}) which is audible to the normal human ear is I_{0}= 10^{-12}W m^{-2}.