Question

Given below are some functions of x and t to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all: (a) y = 2 cos (3x) sin (10t) (b) y = 2 x − υ t (c) y = 3 sin (5x – 0.5t) + 4 cos (5x – 0.5t) (d) y = cos x sin t + cos 2x sin 2t

Answer 1

a)

The general equation of a travelling wave is,

y( x,t )=Asinω( x±vt )

Here, the amplitude of the wave is A, it’s angular velocity is ω, it’s position is x and it’s velocity is v.

The general equation of a stationary wave is,

y( x,t )=Asinωtsin nπx L

Here, the amplitude of the wave is A, it’s angular velocity is ω, it’s position is x and it’s velocity is v.

The given equation is,

y( x,t )=2cos( 3x )sin( 10t )

As the given equation is similar to the standard standing wave equation, so the given wave is a stationary wave.

b)

The general equation of a travelling wave is,

y( x,t )=Asinω( x±vt )

Here, the amplitude of the wave is A, it’s angular velocity is ω, it’s position is x and it’s velocity is v.

The general equation of a stationary wave is,

y( x,t )=Asinωtsin nπx L

Here, the amplitude of the wave is A, it’s angular velocity is ω, it’s position is x and it’s velocity is v.

The given equation is,

y( x,t )=2 x−vt

As the given equation corresponds to neither the standard standing wave equation nor the standard travelling wave equation, so the given wave is neither a stationary wave nor a travelling wave.

c)

The general equation of a travelling harmonic wave is,

y( x,t )=Asinω( x±vt )

Here, the amplitude of the wave is A, it’s angular velocity is ω, it’s position is x and it’s velocity is v.

The general equation of a stationary wave is,

y( x,t )=Asinωtsin nπx L

Here, the amplitude of the wave is A, it’s angular velocity is ω, it’s position is x and it’s velocity is v.

The given equation is,

y( x,t )=3sin( 5x−0.5t )+4cos( 5x−0.5t )

As the given equation is similar to the standard harmonic wave equation, so the provided wave is a travelling harmonic wave.

d)

The general equation of a travelling wave is,

y( x,t )=Asinω( x±vt )

Here, the amplitude of the wave is A, it’s angular velocity is ω, it’s position is x and it’s velocity is v.

The general equation of a stationary wave is,

y( x,t )=Asinωtsin nπx L

Here, the amplitude of the wave is A, it’s angular velocity is ω, it’s position is x and it’s velocity is v.

The given equation is,

y( x,t )=cosxsint+cos2xsin2t = y 1 ( x,t )+ y 2 ( x,t )

As the given equation represents the superposition of the two standard stationary waves, so the provided wave is an equation of the modified stationary wave.

Thus, the given wave equation is the resultant of superposition of two stationary waves.