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Differential Equation of A Wave Motion

Differential Equation of A Wave Motion :- To study wave motion quantitatively and predict its behavior, we use a mathematical relation known as the differential equation of wave motion. This equation forms the backbone of wave theory.

Consider a plane progressive wave travelling in the positive x-direction with velocity , represented by :

$\displaystyle y=Asin(\omega t-kx)$ …..(1)

First partial derivatives of above equation :

$\displaystyle \frac{\partial y}{\partial x}=-kAcos(\omega t-kx)$

$\displaystyle \frac{\partial y}{\partial t}=\omega Acos(\omega t-kx)$

Second partial derivatives :

$\displaystyle \frac{{{\partial }^{2}}y}{\partial {{x}^{2}}}=-{{k}^{2}}Asin(\omega t-kx)$ …..(2)

$\displaystyle \frac{{{\partial }^{2}}y}{\partial {{t}^{2}}}=-{{\omega }^{2}}Asin(\omega t-kx)$ …..(3)

Dividing (2) by (3), we have

$\displaystyle \frac{{}^{{{\partial }^{2}}y}\!\!\diagup\!\!{}_{\partial {{x}^{2}}}\;}{{}^{{{\partial }^{2}}y}\!\!\diagup\!\!{}_{\partial {{t}^{2}}}\;}=\frac{{{k}^{2}}}{{{\omega }^{2}}}=\frac{1}{\left( {}^{{{\omega }^{2}}}\!\!\diagup\!\!{}_{{{k}^{2}}}\; \right)}=\frac{1}{{{\upsilon }^{2}}}$

$\displaystyle \therefore \frac{{{\partial }^{2}}y}{\partial {{x}^{2}}}=\frac{1}{{{\upsilon }^{2}}}\frac{{{\partial }^{2}}y}{\partial {{t}^{2}}}$ …..(4)

This is the differential equation of a wave motion travelling with speed v. The general solution of this equation is of the form :

$\displaystyle y(x,t)=f(ax\pm bt)$ …..(5)

The negative sign corresponds to propagation along the positive x-direction, while the positive sign corresponds to propagation along the negative x-direction.

Significance of the Wave Equation

It is the fundamental condition for any motion to be wave motion.
It proves that waves transfer energy and disturbance, not matter.
Any function of x and t which satisfies equation (4) or which can be written as equation (5) represents a progressive wave. The only condition is that the function must be finite and single-valued at every point of space and at all instants of time. Further, if all these conditions are satisfied, then the speed of the wave is given by :

$\displaystyle \upsilon =\frac{\left| coefficient\text{ }of\text{ }t \right|}{\left| coefficient\text{ }of\text{ }x \right|}=\frac{b}{a}$ …..(6)

Example 1.

Which of the following functions represent waves ? Among them, identify the progressive waves and state their direction of propagation wherever applicable :

(a) $\displaystyle y=Asin\omega t$

(b) $\displaystyle y=A\sin kx$

(d) $\displaystyle y=Asin\left( kx \right)sin\left( \omega t \right)$

(e) $\displaystyle y=Asin(kx-\omega t)+Bcos(kx+\omega t)$

(f) $\displaystyle y=\sqrt{(ax+bt)}$

(g) $\displaystyle y={{(ax-bt)}^{2}}$

(h) $\displaystyle y=A{{e}^{-B{{(x-\upsilon t)}^{2}}}}$

(i) $\displaystyle y=Aco{{s}^{2}}(kx-\omega t)$

(j) $\displaystyle y={{({{x}^{2}}-{{\upsilon }^{2}}{{t}^{2}})}^{2}}$

Solution :

(a) $\displaystyle y=Asin\omega t$

Depends only on time
No propagation

❌ Not a wave.

(b) $\displaystyle y=A\sin kx$

Depends only on position
No motion

❌ Not a wave.

Function of
Shape travels without distortion

✅ Progressive wave
➡ Direction : +x

(d) $\displaystyle y=Asin\left( kx \right)sin\left( \omega t \right)$

Product of space & time terms
Represents nodes and antinodes
Can be written as superposition of two opposite waves

$\displaystyle Asin\left( kx \right)sin\left( \omega t \right)=\frac{A}{2}\left[ cos(kx-\omega t)-cos(kx+\omega t) \right]$

No net propagation

✅ Wave
❌ Not a progressive but it is a standing wave

(e) $\displaystyle y=Asin(kx-\omega t)+Bcos(kx+\omega t)$

Sum of two travelling waves
First term → wave moving +x
Second term → wave moving –x

✅ Wave (superposition principle)

❌ Not a single progressive wave

(f) $\displaystyle y=\sqrt{(ax+bt)}$

Depends only on
Shape moves without changing
No restriction that wave must be sine/cosine
Defined only when (
Not defined for all
Derivative → infinite at (

❌ Not physically acceptable everywhere

📌 Mathematically: satisfies wave equation where defined
📌 Physically: ❌ Rejected

(g) $\displaystyle y={{(ax-bt)}^{2}}$

Depends only on
Entire shape shifts with time

✅ Progressive wave

❓ What about ?

Yes,
But this limit is not physically relevant
No real string, air column, or water surface extends to infinity

📌 So this does NOT violate the wave condition

(h) $\displaystyle y=A{{e}^{-B{{(x-\upsilon t)}^{2}}}}$

Gaussian pulse
Depends only on

✅ Progressive wave (pulse)
➡ Direction: +x

(i) $\displaystyle y=Aco{{s}^{2}}(kx-\omega t)$

Function of
Non-sinusoidal but valid

✅ Progressive wave
➡ Direction: +x

(j) $\displaystyle y={{({{x}^{2}}-{{\upsilon }^{2}}{{t}^{2}})}^{2}}$

and t appear separately
Cannot be written as

❌ Not a wave

Example 2.

Which of the following functions represent a travelling wave ?

(a) $\displaystyle {{\left( x-\upsilon t \right)}^{2}}$

(b) $\displaystyle ln\left( x+\upsilon t \right)$

(d) $\displaystyle \frac{1}{x+\upsilon t}$

Solution :

(a) $\displaystyle {{\left( x-\upsilon t \right)}^{2}}$

As ,
Displacement becomes infinite
Not physically possible

🚫 Rejected as a physical travelling wave

(b) $\displaystyle ln\left( x+\upsilon t \right)$

Undefined for (
Becomes infinite near zero
Not finite at all points

🚫 Rejected

Always finite
Smooth, localized pulse
Goes to zero as

✅ Finite everywhere and physically realizable

✔ Accepted

(d) $\displaystyle \frac{1}{x+\upsilon t}$

Becomes infinite at
Singularity (infinite displacement)

🚫 Rejected

Although all four functions are of the form and hence satisfy the wave equation, only option (c) is finite everywhere at all times and hence represents a physically acceptable travelling wave.

Differential Equation of A Wave Motion

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