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

define sound waves

• 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

“waves and sound “

• 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 diagram “

• 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.

“types of sound “

• 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: \(f_5=5\left(\frac{v}{4 l}\right)=5 f_1\)

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“what is sound vibration “

• 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, 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 (l₀) which is audible to the normal human ear is I₀ = 10^-12W m^-2.