IEMDaily - Video Lecture Notes

Example

Write longitudinal waves as sinusoidal waves of displacement of particles of medium

Example: A longitudinal progressive wave is given by the equation

y = 5 \times 10^{- 2} s i n π (400 t + x)

. Find the amplitude and wave length of the wave. [

y, x

are in

m

].

Solution:

y = A s i n (ω t + k x)

y = 5 \times 10^{- 2} \sin (π t \times 400 + π x)

Comparing, we get

A = 5 \times 10^{- 2}

and

k = π

Now,

λ = \frac{2 π}{k}

or,

λ = 2 m

Example

Write longitudinal waves as sinusoidal waves of pressure variation

Example: The equation of displacement due to a sound wave is

S = S_{0} s i n^{2} (ω t - k x)

. If the bulk modulus of the medium is

B

, find the equation of pressure variation due to that sound.

Solution:
The equation for pressure variation due to sound ( Pressure in a sound wave is equal to product of elasticity of gas with the ratio of particle speed to wave speed):

P = - B \frac{\partial S}{\partial x}

, where

B

is the Bulk modulus or elasticity of the gas.
Given,

S = S_{0} \sin^{2} (ω t - k x)

So,

P = - 2 S_{0} B \sin (ω t - k x) \cos (ω t - k x) (- k)

= (- S_{0} B) \times (- k) \times (2 \times \sin (ω t - k x) \cos (ω t - k x))

= S_{0} B k \sin (2 ω t - 2 k x) = B k S_{0} \sin (2 ω t - 2 k x)

Result

Write longitudinal waves as sinusoidal waves of density variation

For longitudinal wave:
There are three equations associated with any longitudinal wave.

y (x, t), Δ P (x, t),

and

Δ ρ (x, t), y

represents displacement of medium particles from their mean position parallel to direction of wave velocity. From

y (x, t)

equation we can make

Δ P (x, t)

and

Δ ρ (x, t)

equation by using fundamental relations between them,

Δ P = - B \frac{d y}{d x}

and

Δ ρ = - ρ \frac{d y}{d x}

, pressure amplitude

= B A K

and density amplitude

= A K

Δ P (x, t)

and

Δ ρ (x, t)

are in same phase . But

y (x, t)

has a phase difference of

\frac{π}{2}

with rest two equations .
We can conclude:
1. Maximum pressure variation is BAK
2. Maximum density variation is

ρ A K

3. Pressure equation and density equation are in phase

Example

Sound Produced by Humans

The human voice is produced by the vocal chords in the larynx. When air is blown from the lungs through the larynx, the vocal cords vibrate at a frequency and produces sound. When these sound waves passes through our mouth and tongue, its pitch and quality is changed and the sound waves are converted into understandable speaking.

Example

Sound needs a medium for propagation

Sound is a form of energy which is transmitted in the form of waves. The wave consists of vibrations of particles of medium through which it travels. If there is no medium, then vibrations in an object will not travel through it.

Example

Bell Jar Experiment

Bell Jar Experiment is used to demonstrate that sound needs a medium to travel. Place an electrical bell in the bell jar and pump out the air of the sealed bell jar. Turn on the electric bell. The sound produced by the bell is not audible to our ears. This demonstrates that the sound waves cannot travel through vacuum and it needs a material medium for its propagation.

Example

Sound can travel through liquids and solids

Sound can travel through solids, liquids and gases. Some materials (like air, water) easily transmit sound waves through them while some materials (like blanket, curtains) absorb most of the incident sound energy. Speed of sound has inverse relation with the square-root of density of the medium.

Definition

Understand mechanism of sound propagation through solids, liquids and gases

Sound waves travel through a mediums solid, liquid, or gas. Sound wave moves through each of these mediums by vibrating the molecules in the medium. The molecules of the solids are packed very tightly. Molecules of liquids are not packed as tightly as solids. And gases are very loosely packed. The spacing of the molecules enables sound to travel much faster through a solid than a gas. Sound travels about four times faster and farther in water than it does in air. This is why whales can communicate over huge distances in the oceans. Sound waves travel about thirteen times faster in wood than air. They also travel faster on hotter days as the molecules bump into each other more often than when it is cold.

Example

Wave propagation from a point source and spherical wave fronts

Acoustic theory tells us that a point source produces a spherical wave in an ideal isotropic (uniform) medium such as air. Furthermore, the sound from any radiating surface can be computed as the sum of spherical wave contributions from each point on the surface (including any relevant reflections). The Huygens-Fresnel principle explains wave propagation itself as the superposition of spherical waves generated at each point along a wavefront.

Thus, all linear acoustic wave propagation can be seen as a superposition of spherical traveling waves.

Example

Speed of sound in different mediums

Speed of sound is different in different media. It is generally more in solids, less in liquids and least in gases because of more elasticity in solids.

Medium	Speed of sound (m/s)
Air	330
Hydrogen	1270
Water	1450
Copper	3560
Glass	5500

Example

Speed of sound in different mediums

Example: Find the ratio of speed of sound wave in Neon to that in H

_{2}

O vapors at any temperature. Molecular mass of H

_{2}

O is 18 and and Neon is 20.

Solution:

v = \sqrt{\frac{γ R T}{M_{0}}}

So,

v \propto \frac{1}{\sqrt{M_{0}}}

or,

\frac{v_{1}}{v_{2}} = \frac{1 / \sqrt{M_{01}}}{1 / \sqrt{M_{02}}}

= \sqrt{\frac{M_{02}}{M_{01}}}

= \sqrt{\frac{18}{20}}

\frac{v_{1}}{v_{2}} = \frac{3}{\sqrt{10}}

Example

Experimental determination of speed of sound in air

Choose two far places (say A and B) in visibility of each other. Fire gun in air at A and measure time required for sound to reach B using a stop clock. Then speed is given by:

v = \frac{d}{t}

Note: Gunshot can be fired at B with recording of time at A. The two readings may differ by a small amount due to human error. Average of two values will give a better estimate of speed of sound.