JEE Main & Advanced Physics Wave Mechanics Reflection and Refraction of Waves

When waves are incident on a boundary between two media, a part of incident waves returns back into the initial medium (reflection) while the remaining is partly absorbed and partly transmitted into the second medium (refraction)

(1) Rarer and denser medium : A medium is said to be denser (relative to the other) if the speed of wave in this medium is less than the speed of the wave in other medium.

In comparison to air speed of sound is maximum in water, hence water is rarer  medium for sound waves w.r.t. air. But it is not true for light (EM-waves). For light waves water is denser medium w.r.t. air.

(2) In reflection or refraction frequency remains same

(3) For reflection angle of incidence (i) = Angle of reflection (r)

(4) In case of refraction or transmission \[\frac{\sin i}{\sin r'}\,=\,\frac{{{v}_{i}}}{{{v}_{t}}}\]

(5) Boundary conditions : Reflection of a wave pulse from some boundary depends on the nature of the boundary.

(i) Rigid end : When the incident wave reaches a fixed end, it exerts an upward pull on the end, according to Newton's law the fixed end exerts an equal and opposite down ward force on the string. It result an inverted pulse or phase change of \[\pi \]. Crest (C) reflects as trough (T) and vice-versa, Time changes by \[\frac{T}{2}\] and Path changes by \[\frac{\lambda }{2}\]

(ii) Free end : When a wave or pulse is reflected from a free end, then there is no change of phase (as there is no reaction force). Crest (C) reflects as crest (C) and trough (T) reflects as trough (T), Time changes by zero and Path changes by zero.

(iii) Exception : Longitudinal pressure waves suffer no change in phase from rigid end i.e. compression pulse reflects as compression pulse. On the other hand if longitudinal pressure wave reflects from free end, it suffer a phase change of \[\pi \] i.e. compression reflects as rarefaction and vice-versa.

(iv) Effect on different variables : In case of reflection, because medium is same and hence, speed, frequency \[(\omega )\] and wavelength \[\lambda \] (or k) do not changes. On the other hand in case of transmitted wave since medium changes and hence speed, wavelength (or k) changes but frequency \[(\omega )\] remains the same.

(6) Wave in a combination of string

(i) Wave goes from rarer to denser medium

Incident wave \[{{y}_{i}}={{a}_{i}}\sin (\omega \,t-{{k}_{1}}x)\]

Reflected wave \[{{y}_{r}}={{a}_{r}}\sin [\omega \,t-{{k}_{1}}(-x)+\pi ]\]\[=-\,a\sin \,(\omega \,t+{{k}_{1}}x)\]

Transmitted wave\[{{y}_{t}}={{a}_{t}}\sin \,(\omega \,t-{{k}_{2}}x)\]

(ii) Wave goes from denser to rarer medium

Incident wave \[{{y}_{i}}={{a}_{i}}\sin (\omega \,t-{{k}_{1}}x)\]

Reflected wave \[{{y}_{r}}={{a}_{r}}\sin [\omega \,t-{{k}_{1}}(-x)+0]\]\[=\,a\sin \,(\omega \,t+{{k}_{1}}x)\]

Transmitted wave\[{{y}_{t}}={{a}_{t}}\sin \,(\omega \,t-{{k}_{2}}x)\]

(iii) Ratio of amplitudes : It is given as follows

\[\frac{{{a}_{r}}}{{{a}_{i}}}=\frac{{{k}_{1}}-{{k}_{2}}}{{{k}_{1}}+{{k}_{2}}}=\frac{{{v}_{2}}-{{v}_{1}}}{{{v}_{2}}+{{v}_{1}}}\]

and \[\frac{{{a}_{t}}}{{{a}_{i}}}=\frac{2{{k}_{1}}}{{{k}_{1}}+{{k}_{2}}}=\frac{2{{v}_{2}}}{{{v}_{1}}+{{v}_{2}}}\]