Sound Waves Concept Page - 11

Definition

Kundt's tube method in finding speed of sound in air

 
The tube is basically a transparent pipe which is kept horizontal which contains a small amount of a fine powder such as cork dust, talc or Lycopodium. At one end of the tube is a source of sound at a single frequency (a pure tone). A metal rod is used as resonator that he caused to vibrate or 'ring' by rubbing it, modern demonstrations usually use a loudspeaker attached to a generator producing a sine waveform. The other end of the tube is blocked by a piston which is movable which can be used to adjust the length of the tube. The sound generator is turned on and the piston is adjusted until the sound from the tube suddenly gets much louder. This indicates that the tube is at resonance. Which essentially means the length of the round-trip path of the sound waves, from one end of the tube to the other and back again, is a multiple of the wavelength of the sound waves. Therefore the length of the tube is a multiple of half a wavelength. At this point the sound waves in the tube are in the form of standing waves, and the amplitude of vibrations of air are zero at equally spaced intervals along the tube, called the nodes. The powder is caught up in the moving air and settles in little piles or lines at these nodes, because the air is still and quiet there. The distance between the piles is one half wavelength λ$\lambda$/2 of the sound. By measuring the distance between the piles, wavelength of the sound in air can be determined. The frequency f of the sound is known, multiplying it by the wavelength gives the speed of sound c in air:

c=wavelength×f$c=wavelength\times f$

The detailed motion of the powder is actually due to an effect called acoustic streaming caused by the interaction of the sound wave with the boundary layer of air at the surface of the tube

Definition

Problem based on frequency and time period

Frequency of sound coming from speaker is of  280 Hz.Find the time period of the sound wave.
Solution: f=1T$f={\displaystyle \frac{1}{T}}$
              280=1T$280={\displaystyle \frac{1}{T}}$
              T=1280=3.6×10−3 s$T={\displaystyle \frac{1}{280}}=3.6\times {10}^{-3}s$

Definition

Qualitatively understand Doppler effect and identify physical examples

The Doppler effect is the change in frequency of a wave for an observer moving relative to its source. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from an observer. Compared to the emitted frequency, the received frequency is higher during the approach, identical at the instant of passing by, and lower during the recession.
When the source of the waves is moving toward the observer, each successive wave crest is emitted from a position closer to the observer than the previous wave. Therefore, each wave takes slightly less time to reach the observer than the previous wave. Hence, the time between the arrival of successive wave   at the observer is reduced, causing an increase in the frequency. While they are travelling, the distance between successive wave fronts is reduced, so the waves "bunch together". Conversely, if the source of waves is moving away from the observer, each wave is emitted from a position farther from the observer than the previous wave, so the arrival time between successive waves is increased, reducing the frequency. The distance between successive wave fronts is then increased, so the waves "spread out".

Definition

Doppler Effect

When the speeds of source and the receiver relative to the medium are lower than the velocity of waves in the medium, the relationship between observed frequency and emitted frequency is given by:

f=(c+vrc+vs)f0$f=({\displaystyle \frac{c+v_r}{c+v_s}})f_0$
where:
c$c$ is the velocity of waves in the medium;
vr$v_r$ is the velocity of the receiver relative to the medium; positive if the receiver is moving towards the source (and negative in the other direction); 
vs$v_s$ is the velocity of the source relative to the medium; positive if the source is moving away from the receiver (and negative in the other direction).

Definition

Understand the significance of medium in Doppler effect

For waves that propagate in a medium, such as sound waves, the velocity of the observer and of the source are relative to the medium in which the waves are transmitted. The total Doppler effect may therefore result from motion of the source, motion of the observer, or motion of the medium. Each of these effects is analyzed separately. For waves which do not require a medium, such as light or gravity in general relativity, only the relative difference in velocity between the observer and the source needs to be considered.

Result

Mechanism of occurence of Doppler effect for a moving source and a stationary observer

Frequency of sound using Doppler's effect is given by:

f=c+vrc+vsf0$f={\displaystyle \frac{c+v_r}{c+v_s}}{f}_0$
where:
c$c$ is the velocity of waves in the medium; vr$v_r$ is the velocity of the receiver relative to the medium; positive if the receiver is moving towards the source (and negative in the other direction).
In the given case observer is stationary, so vr=0$v_r=0$ 
vs$v_s$ is the velocity of the source relative to the medium; positive if the source is moving away from the receiver (and negative in the other direction).
Ultimately formula narrows down to:
f=cc+vsf0$f={\displaystyle \frac{c}{c+v_s}}{f}_0$

Example

Use the formula for observed frequency for a moving source and a stationary observer

A fire engine with its bell ringing with a frequency of 200 Hz is moving with a velocity of 54 kmph towards an observer at rest near a hut on fire. The apparent frequency of sound heard by the observer is (velocity of sound in air = 300 m/s)Given : V0=0m/s$V_0=0m/s$
Vs=54km/h$V_s=54km/h$
=54×10003600$=54\times {\displaystyle \frac{1000}{3600}}$
=15m/s$=15m/s$
V=300m/s$V=300m/s$
δ=200Hz$\delta =200Hz$
By Doppler effect formula
δ1=δ(V−V0V−Vs)${\delta }^1=\delta ({\displaystyle \frac{V-V_0}{V-Vs}})$
=200(300−0300−15)$=200({\displaystyle \frac{300-0}{300-15}})$
=200×300285$={\displaystyle \frac{200\times 300}{285}}$
=210.5Hz$=210.5Hz$

Result

Understand the mechanism of occurence of Doppler effect for a stationary source and a moving observer

Frequency of sound using Doppler's effect is given by:

f=c+vrc+vsf0$f={\displaystyle \frac{c+v_r}{c+v_s}}{f}_0$
where:
c$c$ is the velocity of waves in the medium; vr$v_r$ is the velocity of the receiver relative to the medium; positive if the receiver is moving towards the source (and negative in the other direction); vs$v_s$ is the velocity of the source relative to the medium; positive if the source is moving away from the receiver (and negative in the other direction).
In the given case source is stationary so vs=0$v_s=0$
Ultimately formula narrows down to:
f=c+vrcf0$f={\displaystyle \frac{c+v_r}{c}}{f}_0$

Example

Derive and use the formula for observed frequency for a stationary source and a moving observer

An observer is moving away  from a sound source of frequency 100Hz. If the observer is moving with a velocity 49m/s and the speed of sound in air is 330m/s, the observed frequency isGiven :V0=49m/s$V_0=49m/s$
δ=100Hz$\delta =100Hz$
V=330m/s$V=330m/s$
Vs=0m/s$V_s=0m/s$
By doppler effect somula 
δ′=δ(V−V0V−Vs)${\delta }^{\prime }=\delta ({\displaystyle \frac{V-V_0}{V-V_s}})$
=100(330−49330)$=100({\displaystyle \frac{330-49}{330}})$
=100×281330$=100\times {\displaystyle \frac{281}{330}}$
=85.15Hz$=85.15Hz$

Result

Understand the mechanism of occurence of Doppler effect for a moving source and a moving observer

For a moving source and a moving observer the formula of frequency according to Doppler's effect is given by:
f=c+vrc+vsf0$f={\displaystyle \frac{c+v_r}{c+v_s}}{f}_0$
where:
c$c$ is the velocity of waves in the medium; vr$v_r$ is the velocity of the receiver relative to the medium; positive if the receiver is moving towards the source (and negative in the other direction); vs$v_s$ is the velocity of the source relative to the medium; positive if the source is moving away from the receiver (and negative in the other direction).

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