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According to Newton, velocity of sound in any medium is given by v =\(\sqrt \frac{E}{ρ}\) where E is the modulus of elasticity and p is the density of the medium.

For gases E = B, bulk modulus

∴ v = \(\sqrt \frac{E}{ρ}\) ……(1)

When sound waves travel through a gas alternate compressions and rarefactions are produced. At the compression region pressure increases and volume decreases and at the rarefaction region pressure decreases and volume increases. Newton assumed that these changes take place under isothermal conditions i.e., at a constant temperature. Under isothermal condition, B = P, pressure of the gas.

∴ In (1) v = \(\sqrt \frac{E}{ρ}\) ………..(2)

This is Newton’s formula for velocity of sound in a gas. 

For air at NTP, P = 101.3 kPa and 

ρ = 1.293 kgm^-3

Substituting the values of P and ρ in . equation (1) we get v = 280m/s. This is much lower than the experimental value of 332 m/s. Thus Newton’s formula is discarded.

Laplace’s correction: According to Laplace, in a compressed region temperature increases and in a rarefied region it decreases and these changes take place rapidly. Since air is an insulator, there is no conduction of heat. Thus changes are not isothermal but adiabatic. Under adiabatic condition, B = γ P, where γ is the ratio of specific heats of the gas.

Substituting in equation (1), v = \(\sqrt \frac{E}{ρ}\) 

The above equation is called Newton- Laplace’s equation Substituting the values of P, ρ and γ in the above equation, gives the velocity of sound in air at NTP to be about 331 m/s. This is in close agreement with the experimental value.

Newton's formula for the velocity of sound in air. Newton was the first to show that velocity of a longitudinal wave (i.e., sound waves) in any medium is given by:

v = √{p/ρ}    ....(i)

Newton assumed that during the compressions and rarefractions, the temperature of the medium does not change. Newton suggested that during compression, the temperature increases and the heat produced is immediately lost to the surrounding. During rarefractions, the heat is lost, which is compansated by gaining heat from the surrondings. Thus, when sound travels in air, conditions are "isothermal". i.e., pressure and volume of the air change. but the temperature remains constant.

Suppose the pressure increases from p to (p + Δp) and the volume decreases from v to (v - Δv). As the temperature does not change, therefore, Boyle's law is applicable.

i.e., pV = (p + Δp)(V - ΔV)

= pV - pΔV + VΔp - Δp.ΔV

Since, Δp and ΔV are very small, so the product Δp.ΔV can be neglected.

Hence, pV = pV - p.ΔV + V.Δp

or pΔV = V.Δp

or p.{p}/{Δv/v} = k, Bulk modulus.

Hence, eqn (i) can be written as

v = √{p/ρ}    ....(ii)

which is the Newton's formula for the velocity of sound in air. Let us calculate the velocity of sound in air at normal temperature and pressure.

At N.T.P we have

p = 0.76 m of Hg.

Now using p = hρg

we get p = 0.76 m x 13,600 kg m x 9.8 ms^-2

= (0.76 x 13600 x 9.8) Nm^-2

and ρ (for air) = 1.293 kg m^-3

Hence, v = √{0.76 x 13600 x 9.8 Nm^-2}/{1.293 Kg m^-3}

= 280 ms^-1

The experimental value of the velocity of the sound at N.T.P comes out to be 332 ms^-1 which is much higher than the theoretical value (i.e., 280 ms-1). This means that there is some error in the Newton's formula. Newton could not give a satisfactory explanation for this error. In 1816, Laplace, a French Scientist explained the discrepency in the Newton's formula as follows:

Laplace's Correction: Laplace pointed out that Newton was wrong in assuming that during compressions and rarefactions, the temperature remains constant. He on the other hand argued that compression and rarefaction follow each other so rapidly that they hardly get time to equalize their temperatures with the surroundings. During compressions there is an increase in temperature whereas during rarefactions the temperature falls. In otherwords, the process during the propagation of sound waves in the medium is adiabatic (where the temperature does not remain constant) and not isothermal as assumed by Newton. Thus Boyle's law is not applicable. For adiabatic process, we have.

pvγ = Consatant    ....(iii)

[where γ = C_p/C_v]

Differentiating equation (iii), we get

pγp^γ-1 dv + v² dp = 0

or γpp^γ-1 dv = v^γdp

or, γp.{v^γdp}/{p^γ-1dv} = {dp}/{-dv/v = k}

Hence, equation (i) becomes,

v = √{γp/ρ}    ....(iv)

For air, the value of γ is 1.41. Therefore, the speed of sound in air at N.T.P. is given by

v = [{1.41 x 76 x 13.6 x 980}/{1.293 x 10^-3}]^-1/2

= 332 ms^-1

This result is in close agreement with the experimental value of the velocity of sound in air. Thus, equation (iv) gives the velocity of sound in the air.

(a) 320 HZ

\(f' = (\frac{v+v_0}{v-v_s}) f\)

⇒ 400 = \([\frac{360+40}{360-40}]\)f

⇒ \(f\) = 320 Hz

Yes, It does not matter whether the sound source or the transmission media are in motion, vibrations will be compressed in the direction of convergence and dilated in the direction of divergence.

Wavelength of sound waves changes.

20 Hz to 20,000 Hz

(a) No wave, (b) longitudinal waves, (c) longitudinal (d) transverse or longitudinal or both (separately), (e) combined longitudinal and transverse ripples.

This occur due to poor quality of sound.

Waves are classified into two types: 

1. Mechanical Waves: Waves, which requires a medium for their propagation are called mechanical waves. e.g.: Waves on the surface of water, Seismic waves (due to earthquake), Sound waves, Waves on stretched string, waves formed in an air column, shock waves. 

2. Non-mechanical Waves: Waves, which do not require a medium for their propagation are called Non-mechanical Waves. e.g.: Light waves, heat waves, radio waves, X- rays, ultraviolet rays, Infrared rays, etc.

Given equation is,

y = 0.2 sin(50t – 0.5x)

= 0.2 sin50(t – x/100)

Comparing with y = a sin ω (t – x/v)

amplitude a = 0.2m,

velocity v = 100m/s

The fundamental frequency for tube A with both ends open is f_A = \(\frac{v}{2L}\)

The fundamental frequency for tube B with one end closed is \(f_B = \frac{v}{4L}\)

\(\frac{f_A}{f_B}\) = \(\frac{v/lL}{v/4L}\) = 2

⇒ f_A : f_B = 2 : 1

(c) 330 \(\sqrt 2\) 

There is no effect of change of pressure on the velocity of sound in air . Further , v ∝ \(\sqrt T\)

Let us assume each disturbance has an amplitude ‘A’ then the resultant displacement is given by,

y = A sin 2π(n – 1)t + A sin2πnt+ A sin2π(n + 1)t

i.e. y= 2A sin 2πnt cos2πt + A sin2πnt

y= A(1 + 2cos2πt) sin2πnt

∴ Resultant amplitude: A(1 + 2 cos2πt)

Amplitude is maximum when cos 2πt = 1

i.e., when 2πt = 2πk k = 0,1,2……….

i.e., when t = 0, 1, 2, 3……….

∴ Time difference between successive maxima = 1 s 

Similarly, amplitude is ‘0’ when cos2πt = -1/2

i.e., when cos 2πt = 2πk + 2π/3

k = 0, 1, 2………….

i.e. when t = k + 1/3

i.e., when t = 1/3, 4/3, 7/3 ……

Again the time difference between successive minima = 1 s 

∴ The frequency of beats is also 1 Hz. 

Thus, one beat is heard per second.

The phenomenon of the change in apparent pitch of sound due to relative motion between the source of sound and the observer is called Doppler effect.  

When two progressive waves of equal amplitude and frequency, traveling in opposite directions along a straight line superpose each other, the resultant wave does not travel in either direction and is called a stationary or standing wave. At some points, the particles of the medium always remains at rest. These are called nodes. At some other points, the amplitude of oscillation is maximum. These are called antinodes.

In a sound wave, a decrease in displacement, i.e., displacement node causes an increase in the pressure, i.e., a pressure antinode is formed. Also, the increase in displacement is due to the decrease in pressure.

(d) pressure change will be maximum at both ends.

pressure change at open ends in zero.

(a) atmospheric pressure

(c) inversly on the square root of density.

(d) directly on the square root of bulk modulus of the medium.

(b) It represents a stationary wave of frequency 60Hz.

(c) It is the result of superposition of two waves of wavelength 3 m, frequency 60Hz each travelling with a speed of 180 m/s in opposite direction.

The density of moist air is less than that of dry air. As the velocity of sound in a gas is inversely proportional to the square root of its density, the velocity of sound in moist air is greater than that in dry air.

When a source of sound moves towards a stationary observer, the increase in pitch is due to actual or apparent change in its wavelength. It is so because due to motion of the source of sound towards observer at rest, waves get compressed as the effective velocity of sound waves relative to source becomes less than the actual. As a result of it, the wavelength of the sound waves decreases and hence the observed pitch increases.

(a) It represents a stationary wave.

(b) It does not represent either a stationary wave or a travelling wave.

(c) It represents a travelling wave.

(d) It is a superposition of two stationary waves.

When the source and the observer are in relative motion with respect to each other and to the medium in which sound propagates, the frequency of the sound wave observed is different from the frequency of the source. This phenomenon is called Doppler Effect.