Equation of Progressive Wave: Progressive wave is one of the types of waves, and it is represented by mathematical equations. Description of a concept of physics is incomplete without involving mathematics in it.  Mathematical equations can represent waves given that they are bounded and single-valued. These wave equations use frequency, velocity and wavelength to represent a wave. Waves of all kinds have fascinated humanity by their sheer beauty and charisma. But then, what are waves?

A wave is a disturbance moving from the source to the surroundings, due to which energy is transferred from one point to another without transferring matter.

What is a Progressive Wave?

Imagine standing on a beach watching one of the waves at a distance that is approaching the shore. What can you observe? The wave is progressing forward. The wave seems to carry energy transmitted from its point of origin to its neighbouring particles. Such waves are called progressive waves.

Characteristics of Progressive Waves

(1) These waves propagate in the forward direction of the medium with a finite velocity.
(2) In progressive waves, energy and momentum are transferred outwards from the wave source, and the wave profile moves in the wave’s propagation direction.
(3) In progressive waves, pressure and density have equal changes at all medium points.
(4) The wave moves with some definite velocity.
(5) When the wave passes through, the particles of the medium vibrate.
(6) The net displacement of the particle is zero.

Equation of Progressive Wave

Progressive waves can be either longitudinal or transverse, but their mathematical representation is the same.  A physical quantity, called a wave function, is needed to represent a wave mathematically. Its value oscillates in space and time about its mean or equilibrium value as the wave propagates through the medium.

For a one dimensional wave along the \( + x\)-direction, the wave function is of the form:

For transverse waves along the stretched string,  can be the displacement of the particle from the equilibrium position.

For sound waves can represent the density or pressure fluctuation from the mean value. For electromagnetic waves (non-mechanical) can represent either the electric field or the magnetic field.

For a stretched string, the origin acts as a source of disturbance, which travels as a transverse wave in the positive direction of the \(x\)-axis with a velocity.

Time measurement starts \(\left( {t = 0} \right)\) when the particle at the origin \(O\) is displaced from its mean position. The displacement y of this particle at the origin at any instant t is represented by \(y = f\left( t \right)\) . Here \(x = 0\).

\(P\) is a point at a distance \(x\) from . The wave reaches \(P\) in \(\frac{x}{v}\) seconds. i.e. the particle displacement \(y\) at any instant will be the same as the displacement of the particle at the origin \(\frac{x}{v}\) seconds earlier.

Hence, the displacement of the particle \(P\) is given by \(y = f\left( {t – \frac{x}{v}} \right)\). This formula is the equation of any shape wave travelling along the positive \(x\)-axis with a constant velocity \(v\). The function determines the exact shape of the wave \(f\).

Similarly, the wave travelling along the negative direction of the \(x\)-axis could be \(y = f\left( {t + \frac{x}{v}} \right)\); hence, the function \(y = f\left( {t \pm \frac{x}{v}} \right)\) represents progressive waves along the \(x\)-axis.

Some Terms Related to Progressive Waves

(i) Wave Number \(\overline n\) : Wave Number is the number of waves present in unit length.

\(\left( {\overline n } \right) = \frac{1}{\lambda }\)

Unit = meter^–1 ;

Dimension \( = {\rm{ }}\left[ {{L^{–1}}} \right]\).

(ii) Propagation constant \((k)\) : \(k = \frac{\varphi }{x} = \frac{{{\rm{phase}}\,{\rm{difference}}\,{\rm{between}}\,{\rm{particle}}}}{{{\rm{Distance}}\,{\rm{between}}\,{\rm{them}}}}h\)
\(k = \frac{\varphi }{v} = \frac{{{\rm{Angula}}\,{\rm{velocity}}}}{{{\rm{Wave}}\,{\rm{velocity}}}}\) and \(k = \frac{{2\pi }}{\lambda } = 2\pi \overline \lambda \)

(iii) Wave velocity (v): The velocity with which the crests and troughs or compression and rarefaction travel in a medium are defined as wave velocity [v = \frac{\omega }{k} = n\lambda = \frac{{\omega \lambda }}{{2\pi }} = \frac{\lambda }{T}\).

(iv) Phase and phase difference: The phase of the wave is given by the sine or cosine argument in the wave equation. It is represented by \(\phi \left( {x,t} \right) = \frac{{2\pi }}{\lambda }\left( {vt – x} \right)\).

At a given position (for a fixed value of \(x\)), phase changes with time \((t)\).

\(\frac{{d\phi }}{{dt}} = \frac{{2\pi V}}{\lambda } \Rightarrow d\phi = \frac{{2\pi }}{T}.dt\) Phase difference \( = \frac{{2\pi }}{T} \times \)

At a given time (for a fixed value of \(t\)), the phase changes with position \((x)\).  \(\frac{{d\phi }}{{dx}} = \frac{{2\pi }}{\lambda } \Rightarrow d\phi = \frac{{2\pi }}{\lambda } \times dx \Rightarrow \) Phase difference \( = \frac{{2\pi }}{\lambda } \times \) Path difference and
Time difference \( = \frac{T}{\lambda } \times \) Path difference

Equation of a Sinusoidal Progressive Wave

Let us consider the example which we had taken earlier, of a stretched string. Let a simple harmonic wave be launched at the point \(O\). Thus, the wave will progressively move forward, and the medium particles will oscillate simply harmonically with an amplitude \(A\), period \(T\) and frequency \(v\).

Let \(\omega \) be the angular frequency. Thus \(\omega = \frac{{2\pi }}{T} = 2\pi v\).

The displacement \(Y\) of the particle at \(O\), at the time \(t\), is given by: \(Y = A\sin \omega t\)

Consider the particle \(P\) at a distance \(x\) from \(O\). The wave profile starting at point \(O\) will be shifted to point \(P\) at a time \(t\) later. So the \(2\) wave profile at a point \(P\) is the same as that of the point \({O^{t – \frac{x}{v}}}\) earlier. The periodic time \(T\) corresponds to a phase shift of \(2\pi \) radian, so the time interval \(t = \frac{x}{v}\) would correspond to a phase shift \(\Delta \phi = \left( {\frac{{2\pi }}{T}} \right)t = \left( {\frac{{2\pi }}{T}} \right)\frac{x}{v}{\rm{radian}}\), which mathematically represents the shifting of the wave profile.

The displacement \(Y\) of the particle at a point \(P\) at a time \(t\) is given by:

\(Y = A\sin \left( {\omega t – \Delta \phi } \right) = A\sin \left( {\frac{{2\pi t}}{T} – \left( {\frac{{2\pi }}{T}} \right)\frac{x}{v}} \right) = A\sin \frac{{2\pi }}{T}\left( {t – \frac{x}{v}} \right)\)

The equations of the sinusoidal progressive wave travelling in the positive direction of \(x\)-axis can also be written as:

 (i) \(y = A\sin \left( {\omega t – kx} \right)\)

(ii) \(y = A\sin \left( {\omega t – \frac{{2\pi }}{\lambda }x} \right)\)

(iii) \(y = A\sin 2\pi \left[ {\frac{t}{T} – \frac{x}{\lambda }} \right]\)

(iv) \(y = A\sin \frac{{2\pi }}{\lambda }\left( {vt – x} \right)\)

(v)  \(y = A\sin \omega \left( {t – \frac{x}{v}} \right)\)

where \(y = \) displacement

  \(A=\) amplitude

 \(\omega = \) angular frequency

             (k=\) propagation constant

             \(T=\) time period

              \(\lambda = \) wave length

              \(v = \) wave velocity

               \(t=\) instantaneous time

               \(x=\) position of particle from origin

Note:

 (a) If the sign between \(t\) and \(x\) terms is negative, the wave propagates along the positive X-axis, and if the sign is positive, the wave moves in a negative \(x\)-axis direction.

(b) The co-efficient of sin or cos functions, i.e. Argument of sin or cos function, i.e. \(\left( {\omega t – kx} \right) = {\rm{phase}}\).

(c) The coefficient \(t\) gives angular frequency \(\omega = 2\pi n = \frac{{2\pi }}{T} = vk\).

(d) The coefficient \(x\) gives propagation constant or wave number \(k = \frac{{2\pi }}{\lambda } = \frac{\omega }{v}\) .

(e) The ratio of the coefficient of \(t\) to that of \(x\) gives wave or phase velocity. i.e., \(v = \frac{\omega }{k}\).

(f) When a given wave passes from one medium to another, its frequency \(v\) does not change.

(g) From \(v = n\lambda \) ; \(v \propto \lambda \) \(\therefore n = {\rm{constant}} \Rightarrow \frac{{{v_1}}}{{{v_2}}} = \frac{{{\lambda _1}}}{{{\lambda _2}}}\)

Solved Problems on the Equation of a Progressive Wave

1) A simple harmonic progressive wave is represented by the equation \(y = 8\sin 2\pi \left( {0.1x – 2t} \right)\) where \(x\) and \(y\) are in \({\rm{cm}}\) and t is in seconds. At any instant, the phase difference between two particles separated by \({\rm{2}}{\rm{.0 cm}}\) in the \(x\)-direction is:

     (a)  \(18^\circ \)                         (b) \(36^\circ \)                   (c) \(54^\circ \)                     (d)  \(72^\circ \)

Solution: (d) \(y = 8\sin 2\pi \left( {\frac{x}{{10}} – 2t} \right)\) given by comparing with standard equation \(y = A\sin 2\pi \left[ {\frac{t}{T} – \frac{x}{\lambda }} \right];\lambda = 10\,{\rm{cm}}\)

So Phase Difference \( = \frac{{2x}}{\lambda }\) path difference \( = \frac{{2\pi }}{{10}} \times 2 = \frac{2}{5} \times {180^ \circ } = {72^ \circ }\)

2) The Equation of a transverse wave travelling on a rope is given by \(y = 10\sin \pi \left( {0.01x – 2\pi t} \right)\) where \(y\) and \(x\) are in cm and t in seconds. The maximum transverse speed of a particle in the rope is about:   

(a) \({\rm{63 cm}}\,{\rm{se}}{{\rm{c}}^{{\rm{ – 1}}}}\)           (b) \({\rm{75 cm}}\,{{\rm{s}}^{{\rm{ – 1}}}}\)          (c)  \({\rm{100 cm}}\,\,{\rm{se}}{{\rm{c}}^{{\rm{ – 1}}}}\)                  (d)  \({\rm{121 cm}}\,\,{\rm{se}}{{\rm{c}}^{{\rm{ – 1}}}}\)

Solution: (a) Standard eq. of travelling wave \(y = A\sin \left( {kx – \omega t} \right)\)

By comparing with the given equation \(A = 10\,{\rm{cm}},\,\,\omega = 2\pi \)

\(A = 10{\rm{ cm}},{\rm{ }}\omega = 2\pi \)

Maximum particle velocity \( = {A_\omega } = 2\pi \times 10 = 63\,{\rm{cm}}\,\,{\rm{se}}{{\rm{c}}^{{\rm{ – 1}}}}\)

3) The displacement \(x\) (in metres) of a particle performing simple harmonic motion is related to time \(t\) (in seconds) as \(x = 0.05\cos \left( {4\pi t + \frac{\pi }{4}} \right)\). The frequency of the motion will be:           

(a) \({\rm{0}}{\rm{.5 Hz}}\)        (b) \({\rm{1}}{\rm{.0 Hz}}\)              (c)  \({\rm{1}}{\rm{.5 Hz}}\)              (d)  \({\rm{2}}{\rm{.0 Hz}}\)

Solution: (d) From the given Equation, coefficient of \(t = \omega = 4\pi \)

\(\therefore n = \frac{\omega }{{2\pi }} = \frac{{4\pi }}{{2\pi }} = {\rm{2Hz}}\)

4) A wave is represented by the equation \(Y = 7\sin \left( {7\pi t – 0.04\pi x + \frac{\pi }{3}} \right)\) \(x\) is in meters and \(t\) is in seconds. The speed of the wave is:                                                                                                   

(a) \({\rm{175 m}}\,{\rm{se}}{{\rm{c}}^{{\rm{ – 1}}}}\)         

(b) \({\rm{49 m}}\,{{\rm{s}}^{{\rm{ – 1}}}}\)         

(c)  \(\frac{{{\rm{49}}}}{\pi }{\rm{ m}}\,{{\rm{s}}^{{\rm{ – 1}}}}\)                

(d)  \({\rm{0}}{\rm{.28}}\,\pi \,{\rm{m}}\,{{\rm{s}}^{{\rm{ – 1}}}}\)

Solution: (a) Standard equation

In a given equation \(\omega = 7x,\,k = 0.04\pi \)

\(v = \frac{\omega }{k} = \frac{{7\pi }}{{.04\pi }} = 175\,{\rm{m}}\,{\rm{se}}{{\rm{c}}^{{\rm{ – 1}}}}\)

5) A wave is represented by the equation \(y = 0.5\sin \left( {10t + x} \right)m\). It is a travelling wave propagating along the \(x\)-direction with velocity.

(a) \({\rm{10 m}}\,{{\rm{s}}^{{\rm{ – 1}}}}\)
(b) \({\rm{20 m}}\,{{\rm{s}}^{{\rm{ – 1}}}}\)                
(c)  \({\rm{5 m}}\,{{\rm{s}}^{{\rm{ – 1}}}}\)                            
(d)  None of these

Solution: (a) \(v = \omega /k = 10/1 = 10\,{\rm{m}}\,{{\rm{s}}^{{\rm{ – 1}}}}\)

6) A transverse progressive wave on a stretched string has a velocity of \(10{\rm{ m}}{{\rm{s}}^{{\rm{–1}}}}\) and a frequency of \({\rm{100 Hz}}\). The phase difference between two particles of the string, which are \({\rm{2}}{\rm{.5 cm}}\) apart, will be:                        

(a) \(\pi /8\)                (b)  \(\pi /4\)                   (c) \(3\pi /8\)               (d) \(\pi /2\)

Solution: (d) \(\lambda = v/n = \frac{{10}}{{100}} = 0.1\,{\rm{m = 10}}\,{\rm{cm}}\)

Phase difference \( = \frac{{2\pi }}{\lambda } \times \) path difference \( = \frac{{2\pi }}{{10}} \times 2.5 = \frac{\pi }{2}\)

Summary

1. A wave is a disturbance moving from the source to the surroundings, due to which energy is transferred from one point to another without transferring matter.
2. Waves, which require a medium for their propagation, are called mechanical waves. 
3. Waves, which do not require any medium for their propagation, are called electromagnetic waves. 
4. Matter waves are waves associated with the fundamental particles that constitute matter.
5. In a progressive wave, energy is transferred outwards from the wave source, and the wave profile moves in the wave’s propagation direction.
6. Progressive waves can be either longitudinal or transverse, but they have the same mathematical representation.
7. A physical quantity, called a wave function, is needed to represent a wave mathematically. Its value oscillates in space and time about its mean or equilibrium value as the wave propagates through the medium.
8. The function \(y = f\left( {t \pm \frac{x}{v}} \right)\) represents progressive waves along the \(x\)-axis, and if it is a sinusoidal progressive wave travelling in the positive direction of the \(x\)-axis, it is represented as \(y = A\sin \omega \left( {t – \frac{x}{v}} \right)\).

FAQs on Equation of a Progressive Wave

Q.1. What is a plane-progressive wave?
Ans: A plane progressive harmonic wave is a wave that travels in given directions without change of its form, and every particle of medium performs simple harmonic motion about their mean position with equal amplitude and period.

Q.2. Is a tsunami a progressive wave?
Ans: Tsunami is long-wavelength, shallow-water, progressive waves caused by the rapid displacement of ocean water. The tsunami is generated by the vertical movement of the earth along faults as seismic sea waves.

Q.3. What are examples of progressive waves?
Ans: Vibration of a string, light waves, water waves are examples of progressive waves.

Q.4. What causes a progressive wave in water?
Ans: The nearly friction-transfer of energy from water particle to water particle in the form of circular paths, or orbits, transmits wave energy across the ocean’s surface and causes the waveform to move. Because the waveform in these waves moves forward, they are known as progressive waves.