Sound wave, displacement equation, pressure equation, intensity, practice problems, FAQs

How do we hear? We hear through waves, sound waves arrive to the outer ear, then travel through the middle and inner ear and transform into meaningful signals sent on to the brain. Our brain uses these signals to organize and communicate with the external world.

Now the question is how our brain interprets the sound waves?

First the sound is converted into electrical signals in the cochlea(inner ear), these signals travel via a complex circuit of auditory nerve pathways to the auditory cortex and other parts of the brain that regulate awareness and sensory perception. (Some of these pathways shut down to let you sleep at night, for example, even if noise is present).

Table of content:

• Sound wave
• Propagation of sound wave
• Displacement equation of sound wave
• Pressure equation of sound wave
• Intensity of sound wave
• Practice problem
• FAQs

Sound wave

A sound wave is a three-dimensional mechanical wave, where the particles of the medium oscillate along the direction of the propagation of the wave. So, the sound wave is a longitudinal wave. It is created by a vibrating source such as a guitar string, human vocal cords, the prongs of a tuning fork, or the diaphragm of a loudspeaker.

All mechanical waves need a medium having the properties of inertia and elasticity for their propagation. Since the sound wave is also a mechanical wave, it also needs a medium like any other mechanical wave.

Sound waves propagate in any medium through a series of periodic compressions and rarefactions of pressure, which are produced by the vibrating source.

Propagation of sound wave

Consider a tuning fork that is producing sound waves. Just before the prongs of the tuning fork set into vibration, the medium is undisturbed .

When the prong moves outward towards the right, it causes disturbance in the medium that compresses the air in front of it, causing the pressure to rise slightly.

The region of increased pressure is known as a compression pulse, and it travels away from the prong with the speed of sound.

After producing the compression pulse, the prong reverses its motion and moves inward. This drags away some air from the region in front of it, causing the pressure to dip slightly below the average pressure. This region of decreased pressure is known as a rarefaction pulse. 

Immediately the compression pulse and the rarefaction pulse travels away from the prong with the speed of sound. This keeps on happening as a result of which sound moves ahead.

Displacement equation of sound wave

A sound wave is nothing but fluctuations of existing parameters of the medium. They are:

(i) Position of the particle

(ii) Localized pressure

As sound propagates in a medium, the particles of the medium execute simple harmonic motion along the direction of propagation of waves, as a result of which the position of the particles of the medium is changing. It is given as follows:

S = S0 sin (kx-t + )

Here, S represents the instantaneous displacement of the xth particle of the medium.

Pressure equation of sound wave

Let us consider a particle at x = x in a medium. As the source (the prongs of the tuning fork) executes SHM, it creates compressions and rarefactions in the medium, which shows that the particles of the mediums also execute SHM along the direction of propagation of waves. The displacement of the xth particle whose initial phase is zero can be written as follows:

Where S represents the instantaneous displacement of the particle.

Let us consider particles at x and x + dx positions of the medium. When the wave propagates,

let the displacements of the particles at x and x + dx be S and S + dS, respectively. The volume enclosed between two particles separated by dx is, V = Adx

The change in volume due to a small displacement dS is,

From the definition of the bulk modulus,

On comparing this equation with, P = P0 cos (t-kx)

We get , P0 = BkS0

Where S0 is excess displacement in the xth position (∆x) of the particles of the medium and P0 is the excess pressure (∆P).

Equation of sound propagation,

The equation of sound wave propagation can be written in the two following ways:

P0 = BkS0 is the excess pressure amplitude.

The displacement and pressure variation in a sound wave propagation differs by a phase difference of 90°.

Figure shows the graph of displacement and pressure with respect to x at t=0 and t=T2.

Intensity of sound wave

Sound waves transfer the energy and the momentum while they propagate through the medium.

The equations of displacement and pressure are,

The power transmitted (P) by the wave across cross section A is,

(Since the average value of cos2 is 12 in one complete cycle.)

Intensity is defined as the power transmitted per unit area. Mathematically, it is defined as follows:

Practice problem

Q 1.The equation of a traveling sound wave along the x-axis is given as S = 6.0sin(600t – 1.8 x), where S is measured in 10-5 m, t in seconds, and x in meters. Find the ratio of the displacement amplitude of the particles to the wavelength of the wave.

Answer: Given,S0= 6 × 10-5 m, = 600 s^-1, k = 1.8 m^-1 (S is measured in 10^-5 m)

Now

Q 2. A sound wave of wavelength 40 cm travels in air. If the difference between the maximum and minimum pressures at a given point is 1.0 × 10-3 Nm-2, then find the amplitude of vibration of the particles of the medium. The bulk modulus of air is 1.4 × 105 Nm-2.

Answer: Given, P_max – P_min = 1.0 × 10^-3 Nm^-2, B = 1.4 × 105 Nm^-2, λ = 40 cm = 0.4 m

The maximum pressure and minimum pressure can be written as:

P_max=P+P₀

P_min=P-P₀

Q 3. The wavelength (λ) of a sound wave traveling in air is 31.4 cm. The displacement of amplitude of individual particles is observed to be 5.5 × 10^-6 m. If the pressure at any point varies between (10⁵ - 55) Pa and (10⁵ + 55) Pa, find the bulk modulus of the medium.

Answer: Given,

Wavelength λ = 31.4 cm

S₀= 5.5 × 10-6 m

P₀=55 Pa

We know that the pressure wave equation is as follows:

P=P₀ cos (wt - kx)

Where,

Q 4. The pressure amplitude in a sound wave from a radio receiver is 2.0 × 10^-2 Nm^-2 and the intensity at a point is 5.0 × 10^-2 Wm^-2. If the pressure amplitude is increased to 2.5 × 10^-2 Nm^-2 by turning the volume knob, then evaluate the intensity.

Answer:

FAQs

Q 1. Can a sound wave go on forever?
Answer: Due to that friction, the wave's amplitude, or height, gets smaller and smaller until it eventually dissipates. That slowly fades out, due to friction in the air. Therefore, to answer the question, sound waves only have a limited amount of time to travel, but yes, in fact they do travel after being emitted.

Q 2. What type of energy does a sound wave have?

a. Mechanical
b. Radiant
c. Electrical
D. Sound

Answer: It has mechanical energy.

Q 3. What will be the effect of temperature on speed of sound?
Answer: The speed of sound increases with the increase of temperature of the medium.

Q 4. Is sound wave needs medium true or false?
Answer: True, any mechanical wave needs medium.