Velocity of traveling wave

Suppose you and your friend are bathing in a swimming pool and you call your friend two times, first when you are out of water and second when you are in water. In which case your friend will reply fastly? When you are in water he will reply faster because the velocity of the traveling sound wave is faster in water than air.

Table of content

• Velocity of transverse wave in string
• Velocity of a longitudinal wave Velocity of longitudinal wave
• Newton’s formula for speed of sound waves in air
• Laplace’s correction
• Velocity of sound in various medium
• Practice problem
• FAQs

Velocity of transverse wave in string

Let us find the velocity of transverse traveling waves on a string. When a jerk is given at one end (left end) of the string, the wave pulses move towards the right end with a velocity v as shown in Figure. Let’s find out the velocity of a transverse traveling wave on a string. When a jerk is given at one of the ends (left end) of the string, the wave pulses move to the right end with a velocity v as shown in the Figure. 

Suppose an observer stands at a distance who is at rest, the pulses move with a velocity v with respect to that observer. Now suppose the observer also moves with the same velocity v in the direction of motion of the wave pulse, then he will notice that the wave pulse is stationary and the rope is moving with a pulse having the same velocity v. Now let’s suppose the observer is also moving with the same velocity v in the direction of motion of the wave pulse, so he will notice that the wave pulse is stationary and the rope is moving with a pulse with the same velocity v.

Let an elemental segment in the string as shown an elemental segment in the string is as shown in Figure below and A and B be two points on the string at an instant of time A and B be the two points on the string at a given instant of time.

Take and be the length and mass of the elemental string, respectively. By definition, linear mass density are the length and mass of the elemental string, respectively. By definition, we know linear mass density, is

The elemental AB has a curvature which is like an arc of a circle with a center at O, radius R and the arc subtending an angle θ at the origin O AB has a curvature which is similar to an arc of a circle with center at O, radius R and the arc subtends an angle θ at the origin O as shown in Figure below. Angle θ can be written as .

The centripetal acceleration due to the tension in the string is

Then, centripetal force Then, the centripetal force will be

Substituting the value of in equation

The tension T in the string acts along the tangent of the elemental segment The tension T in the string, as we know, acts along the tangent to the elemental segment at A and B. Since the arc length of an element is very small at A and B. Since the arc length of any element is very small, change in the tension force can be ignored in the tension can be ignored. On resolving T into horizontal component Tcos(θ/2) and vertical component T sin(θ/2). T into the horizontal component, i.e. Tcos(θ/2) and the vertical component. i.e. T sin(θ/2). 

The horizontal components of tension T at A and B are equal in magnitude but opposite in direction The horizontal components of tension T, at A and B are equal to each other in magnitude but opposite in direction; Hence they cancel each other they cancel each other out. And vertical components will add up. The net radial force is

In comparison, the length of string, amplitude is very small. The sine of a small angle is approximated as . Hence, equation can be written as The sine of any small angle is approximated as . Hence, the equation becomes

But, therefore substituting in equation , we get

Applying Newton’s second law to the element in the radial direction. In equilibrium, the radial component of the force will be equal to the centripetal force. Hence equating equation Applying Newton’s second law on the element in the radial direction. In equilibrium, we know the radial component of the force will be equal to the centripetal force. Hence, equating equation and equation , we have

This formula can be used to find the velocity of a transverse wave in a stretched string where T is tension in string and is linear mass density.

Velocity of a longitudinal wave Velocity of longitudinal wave

Let's take an elastic medium having a fixed mass contained in a cylinder whose cross sectional area is A and having a pressure P. To generate longitudinal waves in the medium, displace the fluid using a piston or keep a tuning fork at one end of the tube.Assuming direction of propagation of waves coincides with the axis of the cylinder. an elastic medium having a fixed mass contained in a cylinder whose cross section is A and has a pressure P. To generate longitudinal waves in the medium, displace the fluid by using a piston or keep a tuning fork at one end of the tube. Let’s Assume direction of propagation of waves coincides with the axis of the cylinder. 

Let ρ be the density of the medium which is initially at rest. At t = 0, the piston at the left end of the tube is set in motion toward the right with Let ρ be the density of the given medium which is initially at rest. At t = 0, the piston at the left end of the tube is brought in motion towards the right with a velocity of u.

Because u is the velocity of the piston and v is the velocity of the elastic wave, so in the time interval ∆t, the distance moved by the piston ∆d = u∆t and the distance moved by theelastic disturbance is ∆x = v∆t. If m is the mass of the air that has a velocity v in a time the velocity of the piston and v be the velocity of the elastic wave, so in time ∆t, the distance moved by the piston ∆d = u∆t and the distance covered by theelastic disturbance is ∆x = v∆t. If m is the mass of the air that has a velocity v in a time interval ∆t . 

Therefore,

Then, the momentum imparted because of motion of the piston with velocity u is

The net impulse is

As we know, the change in momentum is equal to impulse.

When the sound wave passes through the medium, the small element having volume (ΔV) undergoes regular compressions and rarefactions. The change in pressure can also be written as When the sound wave passes through the medium, the small element which has a volume (ΔV) undergoes regular compressions and rarefactions. The change in the pressure can be written as

where, B is known as the bulk modulus of the elastic medium and V is the original volume.

But and

Therefore, 

Equating equation and

This formula can be used to find the velocity of a Longitudinal wave velocity of Longitudinal wave in an elastic medium.

In general velocity of wave a longitudinal wave in elastic medium is

Where E is the modulus of elasticity of medium

For solid :

Speed of sound for solid

where is the Young’s modulus and is the density of material.

For liquids:

Speed of sound for Liquid

where, is the bulk modulus and is the density of the fluid.

Newton’s formula for speed of sound waves in air Newton’s formula for the speed of sound wave in air

According to Newton, when sound propagates in air, the formation of compression and rarefaction takes place in a very slow manner, when sound waves propagate in air, the formation of compression and rarefaction takes place in a very slow way, hence the process is isothermal in nature. That is, the heat produced during compression, and heat lost during rarefaction occur over a period of time so the temperature of the medium remains constant. 

the process is isothermal. So, the heat produced during compression, and heat lost during rarefaction occur over a period of time, so the temperature of the medium will remain constant. 

Treating air as an ideal gas, pressure and volume obey Boyle’s law, Mathematically

Differentiating above equation, we have

where, is the isothermal bulk modulus of air.

Hence the speed of longitudinal wave in air is

Where P is the pressure of air whose value at NTP P is the pressure of the air, whose value at NTP is 76 cm of mercury, we have

and ρ is density of air . 

The speed of sound in air at Normal Temperature and Pressure (NTP)

The speed of sound in air under the conditions of Normal Temperature and Pressure (NTP)

In experiment the speed of sound in air at 0°C is observed as , which is more than 16% of the theoretical value.so the assumption of the isothermal process by Newton is correct according to experimental results.

Laplace’s correction

According to Laplace, when the sound propagates through a medium, the particles oscillate very rapidly such that the compression and rarefaction occur very fast because of this the exchange of heat produced due to compression and cooling due to rarefaction do not take place, because air is a bad conductor of heat. Hence the temperature is not considered as a constant, and sound propagation is an adiabatic process. when the sound propagates through a medium, the particles oscillate very rapidly such that the compression and rarefaction occur very fast because of this the exchange of heat produced due to compression and cooling due to rarefaction do not take place, because air is a bad conductor of heat. Hence the temperature is not considered as a constant, and sound propagation is an adiabatic process. Due to adiabatic process, the gas gas will follow Poisson's law , which is

Here is an adiabatic constant which is the ratio between specific heat at constant pressure and specific heat at constant volume. which is the ratio of the specific heat at constant pressure and the specific heat at constant volume.

Differentiating above equation, we have

where, is the adiabatic bulk modulus of air.

Hence the speed of longitudinal wave in air is

Since air contains mainly, nitrogen, oxygen, hydrogen etc Since the air consists of mainly, nitrogen, oxygen, hydrogen etc so . Hence, the speed of sound in air is 

which isvery much closer to experimental data. which isvery close to experimental data.

Factor affecting speed of sound

1.Effect of temperature

The relation between the speed of sound and temperature is given by,

∝

(For a given gaseous medium, γ, R, and M are fixed)

Thus, the velocity of sound increases with the increase in temperature.

From the given equation,

2. Effect of pressure

If the temperature is constant and P changes, then it changes in such a way that

P, ρ remains constant.

Hence, the pressure does not have any effect on the velocity of sound as long as the temperature is constant.

3. Effect of humidity

With the increase in humidity, the effective density of air decreases, and the velocity of sound increases with the increase in humidity.

Velocity of sound in various medium

The velocity of sound in various mediums at different temperature is given as

Practice problem

Q. A very thin uniform string of length l hangs from a rigid support. Find the speed of the transverse wave on the string at a distance x from the lower end.

A.Suppose that the wave pulse is at the lower end of the string at time t = 0, and the tension in the lower end is zero. Therefore, the velocity of the wave pulse at time t = 0 is zero. 

Now, consider that at any time t, the wave pulse is at a distance x from the lower end of the string. Therefore, the mass of the part of the string up to a distance x from the lower end is μx; hence, the tension in the string at a distance x is, T = μxg

(Recreate)

Thus, the velocity of the wave pulse at this time is, 

Hence, the speed of the transverse wave on the string at a distance x from the lower end is, .

Q. A heavy, uniform rope of length l is suspended from a ceiling. If the rope is given a sudden sideways jerk at the bottom, how long will it take for the pulse to reach the ceiling? 

A.We know that the velocity of the wave pulse at any distance x from the lower end of the string is,

Therefore, the acceleration of the wave pulse is, 

Therefore, the acceleration of the wave pulse is constant.

Now, we know that from the equation of motion in 1-D.

Here, initial velocity of the wave pulse is u=0 and the pulse covers distance l in time

Thus, 

Q. The equation of a progressive wave is , What is the velocity of the wave.

A. Given equation of wave is

Velocity of wave

 .

Q. The speed of longitudinal waves in a solid with young's modulus and density.

A. The speed of longitudinal waves in a solid bas is given by

where is the Young’s modulus and is the density of material.

Ans.

FAQs

Q. Is wave speed in a string dependent upon the frequency of source? 
A. No, wave speed does not depend on frequency.

Q. How does the temperature affect the velocity of sound ?
A. Medium at higher temperatures have more energy and vibrate faster and allow sound waves to travel more quickly.

Q. Why is the velocity of sound more in steel rod than air?
A. In steel, the molecules are closer and thus the atoms are more densely packed than in air.

Q. On which factor velocity of transverse waves in a string depends?
A. Velocity depends on the tension in string and mass per unit length of string.

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