Sound Waves Notes

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: \(f_{\mathrm{P}}=\frac{p}{2 l} \sqrt{\frac{F}{\mu}}=p f_1\)

Sound Waves Synopsis Modes of vibrations in a stretched string

  • 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 M0 = molar mass.
  • Sound Waves Are Pressure Wives: The excess pressure as a function
    of x and t is
    p = p0cos (ωt- kx), where p0 = 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: f1 = \(\frac{v}{4l}\)
  • First Overtone Or Third Harmonic: \(f_3=3\left(\frac{v}{4 l}\right)=3 f_1\)
  • Second Overtone Or Fifth Harmonic: \(f_5=5\left(\frac{v}{4 l}\right)=5 f_1\)

Sound Waves Synopsis Vibration of an air column

  • 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, v0 = velocity of the observer, vs = 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 (l0) which is audible to the normal human ear is I0 = 10-12W m-2.

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