Principle of superposition of waves:

Two or more waves can traverse same path in the given space, independent of one another. This signifies that resultant displacement of the particle at given time is simply algebraic sum of displacements which are given to particle by individual waves. In other words, we can say that resultant displacement of particle is found just by adding algebraically displacements because of individual waves. This is called as superposition of waves.

The interesting case of superposition of waves is that of radio waves. Radio waves of different frequencies are transmitted by different radio stations to broadcast the programs. When they fall on receiving antenna, resultant electric current set up in antenna is fairly complex due to superposition of different waves. However, we can still tune to particular station. That is, out of the many, we can still select and choose particular wave we want. If we have the wave group attained by superposition of the large number of individual waves, we can still separate different waves which were superposed. This is indicative of individual behavior of waves, that is the basis of superposition principle in waves.

To show principle of superposition considers two pulses travelling on rope in opposite directions as shown in figure. Before and after crossing each other, they act completely independently. At the time of crossing, the resultant displacement is the algebraic sum of the individual displacements. It lies in linearity of equation. Consider two waves acting independently on the particle at any position x. Let y₁(x,t) and y₂(x,t) be displacements of particle at the instant of time t because of two waves. Then resultant displacement Y(x, t) of particle is mathematically written as:

Y(x,t) = y1(x,t) + y2(x,t)

that a wave is fundamentally characterized by its angular frequency, amplitude, wave vector and phase. Consider superposition of following pair of waves.

(i) y₁ = a₁sin(ωt - kx) and y₂ = a₂sin(ωt - kx)

(ii) y₁ = asin(ωt - kx) and y₂ = asin(ωt - kx + Φ)

(iii)y₁ = asin(ω₁t - k₁x) and y₂ = asin(ω₂t - k₂x)

(iv)y₁ = asin(ωt - kx) and y₂ = asin(ωt + kx)

Conclusions from above combination of waves are:

(a) In case (i) only the amplitude of two waves differ.

Now let us consider superposition of two waves of same angular frequency, wave vector and phase but different amplitude. These two waves are shown in case (i). Now applying Eq.

we can calculate that the resultant wave is given by

Y(x,t) = a₁sin(ωt - kx) + a₂sin(ωt - kx)

= (a₁+ a₂)sin(ωt - kx)

This equation signifies that resultant wave has same the frequency and phase and resultant amplitude is (a₁ + a₂). It is shown in Figure given below.

(b) In case (ii) only phase of two waves vary. Now consider superposition of two waves that have same amplitude, frequency and wave vector but vary in phase. When such waves superpose, you will find that phenomenon of interference will take place.

(c) In case (iii), frequency ω and wave vector k of two waves vary. Now let us consider case when frequencies and wave vectors of two waves vary slightly. In such a case, irrespective of phase difference superposition results in interesting phenomenon of Beats. If though several waves of slightly different frequencies superpose, then they form the waves group (or wave packet). This gives rise to group velocity, fairly distinct from wave velocity.

(d) In case (iv), waves equations have different signs before wave vector (k). In this situations first wave y1(x, t) is propagating along positive direction of x -axis, while other wave, y2(x, t), is propagating in negative direction along x -axis. This means that they are propagating in opposite directions. When such type of waves superpose then stationary or standing waves are generated.

Stationary waves:

Stationary waves result if two waves of similar angular frequency (i.e., ω) and wavelength i.e. of similar wave vector k), and of similar amplitude travelling in opposite directions superpose on each other. To realize waves of exactly the same amplitude and wavelength, it is simpler to consider one wave as incident wave, and other as reflected wave from the rigid boundary. Reflection of incident wave can occur at fixed boundary like that of string fixed to the wall, or closed end of the organ pipe) or at a free boundary (like free end of the string, or the open end of an organ pipe). At the fixed boundary, displacement y(x, t) remains zero, and reflected wave changes its sign. At the free boundary, though, reflected wave has the similar sign as incident wave. In other words, at the fixed boundary, the phase change of π occurs, while at free boundary, no such change of phase occurs.

Let us consider case where reflection is occurring at free boundary. In this situation, resultant displacement is provided by:

Y(x,t) = asin(ωt - kx) + asin(ωt + kx)  = 2asinωtcoskx

This can be written as:

Y(x,t) = (2acoskx)sinωt

From Equation given above amplitude is given by (2acos kx) that is not fixed. It is dependent (or differs harmonically) on position x of particle. Further, resultant motion has same frequency and wavelength as individual waves. The particles distributed along x -axis executes vibrations perpendicular to x -axis. Amplitudes with which they execute these vibrations are different at different positions along x -axis. Though, time period of vibrations of all particles is same.

Equation doesn't represent travelling wave as argument of sine function is independent of space variable x. Although we began with two waves propagating in opposite directions, we have ended up with something which doesn't propagate in space. Wave which doesn't travel (or propagate) is known as the stationary (or a standing) wave. As it doesn't propagate, it transports no energy along with it. It is clear that displacement Y(x, t) is maximum when

coskx = (cos2π/λ)x = ±1 and minimum when coskx = (cos2π/λ)x = 0

To satisfy equation we need (2π/λ)x = mπ. Similarly Equation needs (2π/λ)x = (2m+1)π/2 with m = 0, 1, 2,.... These provide points of maximum displacement at x= 0, λ/2, λ,...... mλ/2; and minimum displacement at x = λ/4, 3λ/4,....., (2m+1)π/4.

Points of maximum displacement are known as Antinodes, while those of minimum displacement are known as Nodes. Distance between any two consecutive nodes or antinodes is  λ/2, while that between a node and antinode is λ/4. Stationary wave results because of superposition of two identical progressive waves travelling in opposite directions. Result is the non-progressive wave in which disturbance is not handed over from one particle to next. Space (or region) where two waves superpose gets separated in compartments or segments. Each segment ends with points known as nodes where displacement of particles is always zero.

Particles at central points of these segments (known as the antinodes) execute vibrations with maximum amplitude. Particles lying in-between the nodes and antinodes execute vibrations with amplitudes lying in between zero and maximum amplitude.

Velocity of the Particle and Strain at any Point in the Stationary Wave:

The velocity of the particle is stated as rate of change of displacement with respect to time. Velocity of the particle in the stationary wave is computed by differentiating resultant displacement Y(x, t) with respect to time keeping x as constant. If we differentiate equation w.r.t time, we get

Velocity = ∂Y/∂t = 2aωcoskxcosωt

Velocity is maximum when cos kx = ± 1, i.e. at points where x = 0, λ/2, λ, ..., mλ/2. Velocity is minimum (zero) when cos kx = 0, i.e. at points where x = λ/4, 3λ/4,...,2m + 1)λ/4. It signifies that velocity is maximum at antinodes where displacement is also maximum. Velocity is zero at nodes where displacement is zero. At points in between antinodes and nodes, velocity slowly decreases from maximum at antinodes to zero at nodes. The strain on the particle in the stationary wave can be computed by differentiating resultant amplitude i.e. Y(x, t) w.r.t. x keeping t constant. If we differentiate equation w.r.t. x we get strain

∂y/∂x = -2aksinkxsinωt

The strain is maximum for particles at nodes where displacement and velocity are zero. Particles at the nodes are stretched by particles moving in opposite directions. Strain is minimum at antinodes where displacement and velocity are maximum. The particles at antinodes always move along with particles at their sides, not causing much strain on particles at antinodes.

In case of stationary waves, particles get separated in segments like P, Q and R. Particles in one segment always move along in same direction. When particles in segment P move up, those in Q move down. When those in Q move up, ones in P move down. That is, in any two adjacent segments, particles move in opposite directions. All particles in the particular segment reach extreme positions at same time, and also pass through mean positions at same time. All this is possible as all particles have same time period T but have different velocities. Particles that have to cover larger distances have greater velocities. Those which have to cover smaller distances, have smaller velocities.

Velocity is maximum, and when it is zero. Writing Eq.

∂y/∂t = 4πafcos(2π/λ)xcos2πft

= (4πa/T)cos(2π/λ)xcos(2π/T)t

Particle velocity is zero for t = T/4 and 3T/4, and is maximum for t = 0, T/2 and T. Therefore, during each time period, particles of medium have their maximum velocity when they pass through mean position, and have zero velocity when they are at the extreme positions.

Harmonics in Stationary Waves:

All musical instruments based on strings utilise the stationary wave phenomenon. A string clamped at both ends allows stationary waves with some fixed wavelengths. If the length of the string is L , the wavelength of the possible stationary waves on this string, starting from the longest wavelength are: 2L, 2/3L, L/2..., etc.

These wavelengths determine the frequencies of oscillation of the string through the relation λf = v. Here v the velocity of the transverse wave on the string. It is given by the relation v = √T/μ where T is the tension in the string, and μ is the linear mass density (mass per unit length) of the string.

The lowest frequency f0 of vibration is called the fundamental frequency. It is given by:

f₀ = v/λ = 1/2L√T/μ

The other frequencies are called the overtones, and are integral multiples of the fundamental frequency f₀. The fundamental frequency is also called first harmonic. The first overtone, with frequency f = 2f₀, is called the second harmonic. The second overtone, with frequency f = f₀, is called the third harmonic, and so on.

Properties of Stationary Waves:

Properties of stationary waves are given below:

(i) Stationary waves are not progressive. In these disturbance is not handed over from one particle to next.

(ii) Amplitude of each particle is not same. It is maximum at antinodes and zero at nodes. In between, it slowly decreases from that at antinode to one at the node, i.e., zero.

(iii) Distance between two consecutive nodes or two consecutive antinodes is half the wavelength of the stationary wave. Medium splits in segments, with length of each segment equal to half the wavelength.

Wave groups and group velocity:

When two waves of slightly different angular frequencies ω1 and ω2 travelling in the same direction, superpose on each other (Case iii). To avoid unnecessary mathematical complexities, we take the amplitudes of the two waves to be equal. The superposition of such two waves is given by

Y(x,t) = asin(ω₁t - kx) + asin(ω₂t - kx)

= 2asin([(ω₁ + ω₂)t - (k₁ + k₂)x]/2)cos([(ω₁ - ω₂)t - (k₁ - k₂)x]/2)

If ω₁ and ω₂, and similarly k₁ and k₂, are only somewhat different, we can write ω₁ - ω₂ = Δω and k₁ - k₂ = Δk

Superposition, results in formation of groups (or segments) known as wave groups (or wave packets). The wave group can travel with the velocity that may be different from that of individual waves, or of the resultant wave. The velocity of the wave group is known as group velocity. Ratio of angular frequency and wave vector of slowly moving part of superposed wave is known as group velocity. It is provided by the given relation

v_g = (Δω/2)/(Δk/2) = Δω/Δk

If group comprises of the number of component waves with angular frequencies lying between ω₁ and ω₂ (with ω₁ ≈ ω₂), and similarly in wave vector k₁ and k₂ (with |k₁| > |k₂|), the group velocity g v is then written as:

v_g = Δω/Δk = dω/dk

The individual wave velocities are equal, i.e.

ω₁/k₁ = ω₂/k₂ = v

Then v_p = (ω₁ + ω₂)/(k₁ + k₂) = (k₁v + k₂v)/(k₁ + k₂) = v and

v_g = ((ω₁ - ω₂)/2)/((k₁ + k₂)/2) = (vk₁ - vk₂)/(k₁ - k₂) = v

i.e., the group velocity is equal to the phase velocity. This is another relation connecting phase and group velocities. The wavelength of the resultant wave is given by

λ = 2π/k

To show difference between phase and group velocities, consider striking example of waves in deep water - known as-gravity waves. These waves are strongly dispersive. For them the phase velocity is found to be proportional to square root of wavelength, i.e.,

v_p = Cλ₁/2 or

v_p = C₁k₁/2

Here, new constant, C₁ = C√2π.vp = C₁k₁/2, therefore, ω = C₁k₁/2. Differentiating ω with respect to k , we get, vg = dω/dk = 1/2C₁k^-1/2 = 1/2v_p.

That is, group velocity for gravity waves is just half of the phase velocity. In other words, for these waves, component wave crests move faster through group as a whole.

Beats:

Beats are frequently observed between two vibrations with equal frequencies. Beat frequency equals difference between frequencies of beating signals. When sound signals interfere, beat signal can at times be heard as the separate note: Tartini tone. Both are helpful and significant in practice for estimating frequencies and for tuning musical instruments.

Beats are loud sounds that we hear at regular intervals of time depending on difference in frequencies of two superposing waves. Beats are frequently utilized by musicians for tuning their instruments.

In terms of frequencies f₁ and f₂, beat frequency is Δf = f₁ - f₂ = Δω/2π. Time elapsed between any two consecutive beats, known as beat period = 1/Δf.

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