Reflection of Waves

Wave reflection happens when a wave hits a surface and bounces back instead of passing through. This is seen in everyday things like echoes, light bouncing off mirrors, or ripples hitting the edge of water. According to the law of reflection, the wave reflects at the same angle it hits the surface. This simple rule is behind useful tools like mirrors, sonar, and radar, helping us in everything from seeing our reflection to detecting objects underwater.

1.0Stationary/Standing Waves

When two identical waves (transverse or longitudinal) propagating in opposite direction superimpose in bounded medium then the resultant wave is called stationary wave or
standing wave. It is of two types -

(1) Transverse stationary waves

(2) Longitudinal stationary waves

2.0Reflection From Rigid End

When a traveling wave reaches a boundary, some or all of it is reflected.For a wave pulse on a string fixed at one end:

The pulse reflects back upon reaching the fixed boundary.
Ideally, no part of the wave is transmitted into the wall.
The reflected pulse is inverted (i.e., its phase is reversed).

Mathematical Analysis

(a) Rigid End:In such a type of reflection incident and reflected waves have phase difference of and direction of propagation are opposite.

$y_{i} = a sin (ω t - k x)$

Note:The direction of propagation reverses. Also, a phase difference of .

$y_{r} = a sin (ω t + k x + π) = - a sin (ω t + k x)$

$y_{res} = y_{i} + y_{r}$

$y_{res} = a [sin (ω t - k x) - sin (ω t + k x)]$

${sin A - sin B = 2 sin (\frac{A - B}{2}) cos (\frac{A + B}{2})}$

$y_{res} = - 2 a sin k x cos ω t$

$- 2 a sin k x \to Amplitude$

$Let A = 2 a sin k x$

$so at x = 0, A = 0$

3.0Reflection From Free End

When a wave pulse reaches the end of a string free to move vertically:

The pulse is reflected, but not inverted.
The free end moves in the same direction as the incoming pulse.
The pulse exerts a force on the free end, causing it to accelerate upward.

Mathematical Analysis

Free End: In such a type of reflection incident and reflected waves are in phase and direction of propagation are opposite.

$y_{i} = a sin (ω t - k x)$

Note:The direction of propagation reverses. But no phase difference.

$y_{r} = a sin (ω t + k x)$

$y_{res} = y_{i} + y_{r}$

$y_{res} = a [sin (ω t - k x) + sin (ω t + k x)]$

${sin A + sin B = 2 sin (\frac{A + B}{2}) cos (\frac{A - B}{2})}$

$y_{res} = 2 a cos k x sin ω t$

$2 a cos k x \to A m pl i t u d e$

Let $A = 2 a cos k x$

so at x = 0, A = 2a

Illustration-1.An incident wave equation $y = A sin (k x - ω t)$ , if at x = 0, node is formed, then find the equation of reflected wave?

Solution:At x = 0, node is formed therefore it is a rigid end.

For incident wave $(y_{i}) = A sin [- (ω t - k x)] \Rightarrow y_{i} = - A sin [ω t - k x]$

For Reflected Wave

$(y_{r}) = - A sin [ω t + k x + π] \Rightarrow y_{r} = - A [- sin (ω t + k x)] = A sin (ω t + k x)$

Illustration-2.For a stationary wave equation $y = 20 cos (\frac{π}{24} x) cos (16 π t)$ where x and y in cm and t in second.Find?

Type of Reflector
Equation of incident and reflected wave

Solution:Compare the given equation with standard equation $y = 2 A cos k x cos ω t 2 A cos k x = 20 cos (\frac{π}{24} x) an d ω = 16 π$ .

At x = 0 amplitude is maximum therefore it represents Free end
By comparing $A = 10 cm, ω = 16 π, k = \frac{π}{24}$ .

$y_{i} = A cos (ω t - k x) y_{r} = A cos (ω t + k x)$

$y_{i} = 10 cos (16 π t - \frac{π}{24} x) y_{r} = 10 cos (16 π t + \frac{π}{24} x)$

Reflection of Waves

Wave reflection happens when a wave hits a surface and bounces back instead of passing through. This is seen in everyday things like echoes, light bouncing off mirrors, or ripples hitting the edge of water. According to the law of reflection, the wave reflects at the same angle it hits the surface. This simple rule is behind useful tools like mirrors, sonar, and radar, helping us in everything from seeing our reflection to detecting objects underwater.

1.0Stationary/Standing Waves

When two identical waves (transverse or longitudinal) propagating in opposite direction superimpose in bounded medium then the resultant wave is called stationary wave or
standing wave. It is of two types -

(1) Transverse stationary waves

(2) Longitudinal stationary waves

2.0Reflection From Rigid End

When a traveling wave reaches a boundary, some or all of it is reflected.For a wave pulse on a string fixed at one end:

The pulse reflects back upon reaching the fixed boundary.
Ideally, no part of the wave is transmitted into the wall.
The reflected pulse is inverted (i.e., its phase is reversed).

Mathematical Analysis

(a) Rigid End:In such a type of reflection incident and reflected waves have phase difference of and direction of propagation are opposite.

$y_{i} = a sin (ω t - k x)$

Note:The direction of propagation reverses. Also, a phase difference of .

$y_{r} = a sin (ω t + k x + π) = - a sin (ω t + k x)$

$y_{res} = y_{i} + y_{r}$

$y_{res} = a [sin (ω t - k x) - sin (ω t + k x)]$

${sin A - sin B = 2 sin (\frac{A - B}{2}) cos (\frac{A + B}{2})}$

$y_{res} = - 2 a sin k x cos ω t$

$- 2 a sin k x \to Amplitude$

$Let A = 2 a sin k x$

$so at x = 0, A = 0$

3.0Reflection From Free End

When a wave pulse reaches the end of a string free to move vertically:

The pulse is reflected, but not inverted.
The free end moves in the same direction as the incoming pulse.
The pulse exerts a force on the free end, causing it to accelerate upward.

Mathematical Analysis

Free End: In such a type of reflection incident and reflected waves are in phase and direction of propagation are opposite.

$y_{i} = a sin (ω t - k x)$

Note:The direction of propagation reverses. But no phase difference.

$y_{r} = a sin (ω t + k x)$

$y_{res} = y_{i} + y_{r}$

$y_{res} = a [sin (ω t - k x) + sin (ω t + k x)]$

${sin A + sin B = 2 sin (\frac{A + B}{2}) cos (\frac{A - B}{2})}$

$y_{res} = 2 a cos k x sin ω t$

$2 a cos k x \to A m pl i t u d e$

Let $A = 2 a cos k x$

so at x = 0, A = 2a

Illustration-1.An incident wave equation $y = A sin (k x - ω t)$ , if at x = 0, node is formed, then find the equation of reflected wave?

Solution:At x = 0, node is formed therefore it is a rigid end.

For incident wave $(y_{i}) = A sin [- (ω t - k x)] \Rightarrow y_{i} = - A sin [ω t - k x]$

For Reflected Wave

$(y_{r}) = - A sin [ω t + k x + π] \Rightarrow y_{r} = - A [- sin (ω t + k x)] = A sin (ω t + k x)$

Illustration-2.For a stationary wave equation $y = 20 cos (\frac{π}{24} x) cos (16 π t)$ where x and y in cm and t in second.Find?

Type of Reflector
Equation of incident and reflected wave

Solution:Compare the given equation with standard equation $y = 2 A cos k x cos ω t 2 A cos k x = 20 cos (\frac{π}{24} x) an d ω = 16 π$ .

At x = 0 amplitude is maximum therefore it represents Free end
By comparing $A = 10 cm, ω = 16 π, k = \frac{π}{24}$ .

$y_{i} = A cos (ω t - k x) y_{r} = A cos (ω t + k x)$

$y_{i} = 10 cos (16 π t - \frac{π}{24} x) y_{r} = 10 cos (16 π t + \frac{π}{24} x)$

Frequently Asked Questions

Why does a wave invert upon reflection at a fixed boundary but not at a free boundary?

Is energy conserved during wave reflection?

How does reflection differ for transverse and longitudinal waves?

How is reflection of waves utilized in modern technology?

Why does reflection happen?

Join ALLEN!

Reflection of Waves

1.0Stationary/Standing Waves

2.0Reflection From Rigid End

3.0Reflection From Free End

Table of Contents

Frequently Asked Questions

Why does a wave invert upon reflection at a fixed boundary but not at a free boundary?

Is energy conserved during wave reflection?

How does reflection differ for transverse and longitudinal waves?

How is reflection of waves utilized in modern technology?

Why does reflection happen?

Join ALLEN!

Reflection of Waves

1.0Stationary/Standing Waves

2.0Reflection From Rigid End

3.0Reflection From Free End

Table of Contents