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The Speed of a Traveling Wave

Have you ever noticed how fast sound travels? Sound and light form traveling waves. Can the velocity of a traveling wave be measured? If yes, on what factors does the velocity of a traveling wave depend? You know about light waves, sound waves and more! How fast are these waves? Let us find out the speed of a traveling wave.

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The Speed of a Traveling wave

A traveling wave is a wave that moves in space. So, what do we mean by a wave velocity or speed? A wave can be represented graphically as shown below. Consider a wave that is traveling in the positive direction of X-axis. We can represent this wave by the equation y = A sin (kx – ωt) or the wave equation. Here, A is the amplitude, and k is the propagation constant.

Traveling wave

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Wave velocity or Phase Velocity

The distance covered by a wave in the direction of its propagation per unit time is called the wave velocity. Let us try and find it for a wave traveling in the positive X direction. Suppose, y = A sin (kx – ωt) represents our traveling wave. Let us say that this wave does not change its form while traveling through any medium. In other words, it maintains its form.

Travelling wave

On the waveform, it retains its displacement as it moves. For a fixed point on the waveform, we must have a constant argument. Hence, we have

Travelling wave

Therefore, differentiating, both side w.r.t., time t, we get

Travelling wave

But dx/dt = wave velocity – v, therefore

Travelling wave

This represents the wave velocity or the phase velocity of a traveling wave. Let us see some special cases now.

Speed of a Transverse Wave on Stretched String

The speed of transverse waves on a string depends on two main factors

  • The linear mass density or mass per unit length, μ (say)
  • The tension T of the string.

Let us look at these terms in a little more detail.

The linear mass density or μ

It is defined as the mass of the string divided by its length l. Therefore, its dimension is [ML-1]. The tension T has the dimension of force — namely, [M L T-2]. Our goal is to combine μ and in such a way as to generate [dimension (L T-2)]. Therefore, if v depends only on T and μ, the relation between them must be:

Travelling wave

Here, C is a dimensionless constant. Hence, the speed of transverse waves on a stretched string is given by:

Travelling wave

Hence, the speed of a wave along a stretched ideal string depends only on the tension and the linear mass density of the string. It is independent of the frequency of the wave.

Speed of a Longitudinal Wave Speed of Sound

In a longitudinal wave, the constituents of the medium oscillate back and forth in the direction of propagation of the wave. We know that the sound waves travel in the form of compression and rarefactions of the small volume element of the medium. Therefore, the property that determines the extent to which the volume of an element of a medium change when the pressure on it changes, is the bulk modulus B, given by:

Travelling wave

Here, ∆V/V is the fractional change in volume produced by a change in pressure ∆P. The SI unit for pressure is N m-2 or pascal (Pa). The dimension of density is [ML-3]. Hence, the dimension of the ratio B/ρ is [LT-2]. Therefore, based on dimensional analysis, the most appropriate expression for the speed of longitudinal waves in a medium is:

Travelling wave

where C is a dimensionless constant and can be shown to be unity. Hence, the speed of longitudinal waves in a medium is given by:

Travelling waveThe speed of propagation of a longitudinal wave in a fluid, therefore, depends only on the bulk modulus and the density of the medium. It can be shown that the speed of a longitudinal wave in the bar is given by,

Travelling wave

The relation between pressure P and volume V is:

PV = NkBT

Where N is the number of molecules in volume V, kB is the Boltzmann constant and T the temperature of the gas (in Kelvin). Therefore, for an isothermal change, it follows from above equation that:

Travelling wave

Hence, substituting in the equation of the speed of longitudinal wave, we get

Travelling wave

This relation was first given by Newton and therefore it is known as Newton’s formula for the speed of sound.

Solved Examples For You

Q 1: Sound velocity is maximum in:

A) H2     B) N2      C) He      D) O2

Solution: A) As we can see from Newton’s formula for the speed of sound, the speed is inversely proportional to the density of the medium. Therefore, from the given options, the molecular mass of Hydrogen is smallest and hence it will have the least density for an equal volume of all the gases. Hence, the speed of sound will be maximum in H2.

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