Question

Explain Reflection of string and sound waves both from a denser and rarer medium

Answer 1

Explanation of Reflection of string and sound waves both from a denser and rarer medium

Answer 2

Explanation of reflection for string waves (transverse) and sound waves (longitudinal) from denser and rarer media:

String Waves (Transverse Waves):

1. Reflection from a Denser Medium:

   • A denser medium boundary for a string is analogous to a fixed end.
   • When a transverse wave (crest or trough) reaches a fixed end, the string is constrained and cannot move.
   • According to Newton's third law, the fixed support exerts a reaction force on the string, which is opposite to the force exerted by the string on the support. This reaction force generates the reflected wave.
   • A crest arriving at a fixed end pulls the string upwards, so the support pulls downwards, creating a reflected trough. Similarly, a trough is reflected as a crest.
   • This means the reflected wave is inverted compared to the incident wave.
   • The phase of the wave changes by $\pi$ radians or $180^\circ$ upon reflection from a denser medium (fixed end).

2. Reflection from a Rarer Medium:

   • A rarer medium boundary for a string is analogous to a free end.
   • When a transverse wave reaches a free end, the end of the string is free to move transversely.
   • A crest arriving at a free end reaches its maximum displacement and is reflected back as a crest. Similarly, a trough is reflected as a trough.
   • The reflected wave has the same orientation as the incident wave.
   • The phase of the wave does not change upon reflection from a rarer medium (free end). The phase change is $0$ radians or $0^\circ$.

Sound Waves (Longitudinal Waves):

Sound waves are longitudinal waves, consisting of compressions (regions of high pressure and density) and rarefactions (regions of low pressure and density). Reflection can be described in terms of displacement or pressure. The phase change upon reflection is $\pi$ for displacement where it is $0$ for pressure, and vice-versa. It is usually more convenient to describe the reflection of compressions and rarefactions in terms of pressure.

1. Reflection from a Denser Medium:

   • A denser medium (e.g., air reflecting from a rigid wall) offers greater resistance to the motion of particles.
   • Consider the pressure wave. A compression is a region of maximum pressure (above equilibrium). When a compression reaches a rigid boundary, the particles are forced against the boundary, leading to a further increase in pressure at the boundary. This pressure maximum is reflected back as a pressure maximum. A rarefaction (pressure minimum) is reflected back as a rarefaction.
   • In terms of pressure, a compression is reflected as a compression, and a rarefaction is reflected as a rarefaction.
   • The phase of the pressure wave does not change upon reflection from a denser medium. The phase change is $0$ radians or $0^\circ$.
   • In terms of displacement, the boundary is like a fixed end where displacement must be zero. A displacement wave reflects from a denser medium with a phase change of $\pi$ ($180^\circ$).

2. Reflection from a Rarer Medium:

   • A rarer medium (e.g., air reflecting from an open end of a pipe) offers less resistance to the motion of particles.
   • Consider the pressure wave. At an open end, the pressure is essentially equal to the atmospheric pressure, meaning the pressure variation is zero.
   • When a compression (pressure maximum) reaches a rarer medium boundary (like an open end), it is reflected as a rarefaction (pressure minimum) to ensure the total pressure variation at the boundary is zero. Similarly, a rarefaction is reflected as a compression.
   • In terms of pressure, a compression is reflected as a rarefaction, and a rarefaction is reflected as a compression.
   • The phase of the pressure wave changes by $\pi$ radians or $180^\circ$ upon reflection from a rarer medium.
   • In terms of displacement, the boundary is like a free end where displacement is maximum. A displacement wave reflects from a rarer medium with a phase change of $0$ ($0^\circ$).

Summary of Phase Changes (relative to the incident wave):

Wave Type	Property Considered	Reflection from Denser Medium	Reflection from Rarer Medium
String	Displacement	$\pi$ ($180^\circ$)	$0$ ($0^\circ$)
Sound	Pressure	$0$ ($0^\circ$)	$\pi$ ($180^\circ$)
Sound	Displacement	$\pi$ ($180^\circ$)	$0$ ($0^\circ$)

The description of compression/rarefaction reflection for sound waves usually refers to the pressure wave behavior.