JEE Main & Advanced Physics Wave Mechanics Common Cases in Doppler's Effect

Case 1 : Source is moving but observer at rest.

(1) Source is moving towards the observer

Apparent frequency \[n'=n\,\left[ \frac{v-0}{v-(+{{v}_{S}})} \right]=n\,\left( \frac{v}{v-{{v}_{S}}} \right)\]

Apparent wavelength \[\lambda '=\lambda \,\left( \frac{v-{{v}_{S}}}{v} \right)\]

(2) Source is moving away from the observer.

Apparent frequency \[n'=n\,\left[ \frac{v-0}{v-(-{{v}_{S}})} \right]=n\,\left( \frac{v}{v+{{v}_{S}}} \right)\]

Apparent wavelength \[\lambda '=\lambda \,\left( \frac{v+{{v}_{S}}}{v} \right)\]

Case 2: Source is at rest but observer is moving.

(1) Observer is moving towards the source.

Apparent frequency \[n'=n\,\left[ \frac{v-\,(-\,{{v}_{O}})}{v-0} \right]=n\,\left[ \frac{v+{{v}_{O}}}{v} \right]\]

Apparent wavelength \[\lambda '=\frac{(v+{{v}_{O}})}{n'}=\frac{(v+{{v}_{O}})}{n\frac{(v+{{v}_{O}})}{v}}=\frac{v}{n}=\lambda \]

(2) Observer is moving away from the source

Apparent frequency \[n'=n\,\left[ \frac{v-(+{{v}_{O}})}{v-0} \right]=n\,\left[ \frac{v-{{v}_{O}}}{v} \right]\]

Apparent wavelength \[\lambda '=\lambda \]

Case 3: When source and observer both are moving

(1) When both are moving towards each other

(i) Apparent frequency \[n'=n\,\left[ \frac{v-\,(-\,{{v}_{O}})}{v-\,(+{{v}_{S}})} \right]=n\,\left[ \frac{v+{{v}_{O}}}{v-{{v}_{S}}} \right]\]

(ii) Apparent wavelength \[\lambda '=\lambda \,\left( \frac{v-{{v}_{S}}}{v} \right)\]

(iii) Velocity of wave with respect to observer \[=(v+{{v}_{O}})\]

(2) When both are moving away from each other.

(i) Apparent frequency \[n'=n\,\left[ \frac{v-\,(+\,{{v}_{O}})}{v-\,(-\,{{v}_{S}})} \right]=n\,\left[ \frac{v-{{v}_{O}}}{v+{{v}_{S}}} \right]\]

\[(n'<n)\]

(ii) Apparent wavelength \[\lambda '=\lambda \,\left( \frac{v+{{v}_{S}}}{v} \right)\] \[(\lambda '>\lambda )\]

Velocity of waves with respect to observer \[=(v{{v}_{O}})\]

(3) When source is moving behind observer

(i) Apparent frequency \[n'=n\,\left( \frac{v-{{v}_{O}}}{v-{{v}_{S}}} \right)\]

(a) If \[{{v}_{O}}<{{v}_{S}},\] then \[n'>n\]

(b) If \[{{v}_{O}}>{{v}_{S}}\] then \[n'<n\]

(c) If \[{{v}_{O}}={{v}_{S}}\] then \[n'<n\]

(ii) Apparent wavelength \[\lambda '=\lambda \,\left( \frac{v-{{v}_{S}}}{v} \right)\]

(iii) Velocity of waves with respect to observer = \[(v-{{v}_{O}})\]

(4) When observer is moving behind the source

(i) Apparent frequency \[n'=n\,\left( \frac{v-\,(-{{v}_{O}})}{v-(-{{v}_{S}})} \right)\]

(a) If \[{{v}_{O}}>{{v}_{S}},\] then \[n'>n\]

(b) If \[{{v}_{O}}<{{v}_{S}}\] then \[n'<n\]

(c) If \[{{v}_{O}}={{v}_{S}}\] then \[n'<n\]

(ii) Apparent wavelength \[\lambda '=\lambda \,\left( \frac{v+{{v}_{S}}}{v} \right)\]

(iii) The velocity of waves with respect to observer = \[(v-{{v}_{O}})\]

Case 4: Crossing (1) Moving sound source crosses a stationary observer

Apparent frequency before crossing

\[n{{'}_{Before}}=\,n\,\left[ \frac{v-0}{v-\,(+{{v}_{S}})} \right]=n\,\left[ \frac{v}{v-{{v}_{S}}} \right]\]

Apparent frequency

\[n{{'}_{After}}=\,n\,\left[ \frac{v-0}{v-\,(-{{v}_{S}})} \right]=n\,\left[ \frac{v}{v+{{v}_{S}}} \right]\]

Ratio of two frequency \[\frac{n{{'}_{Before}}}{n{{'}_{After}}}=\,\left[ \frac{v+{{v}_{S}}}{v-{{v}_{S}}} \right]>1\]

Change in apparent frequency \[n{{'}_{Before}}-n{{'}_{After}}=\frac{2n{{v}_{S}}v}{({{v}^{2}}-v_{S}^{2})}\]

If \[{{v}_{S}}<<v\] then \[n{{'}_{Before}}-n{{'}_{After}}=\frac{2n{{v}_{S}}}{v}\]

(2) Moving observer crosses a stationary source

Apparent frequency before crossing

\[n{{'}_{Before}}=\,n\,\left[ \frac{v-\,(-\,{{v}_{O}})}{v-0} \right]=n\,\left[ \frac{v+{{v}_{O}}}{v} \right]\]

Apparent frequency

\[n{{'}_{After}}=\,n\,\left[ \frac{v-(+{{v}_{O}})}{v-\,0} \right]=n\,\left[ \frac{v-{{v}_{O}}}{v} \right]\]

Ratio of two frequency \[\frac{n{{'}_{Before}}}{n{{'}_{After}}}=\,\left[ \frac{v+{{v}_{S}}}{v-{{v}_{S}}} \right]\]

Change in apparent frequency \[n{{'}_{Before}}-n{{'}_{After}}=\frac{2n{{v}_{O}}}{v}\]

Case 5: Both moves in the same direction with same velocity \[n'=n,\], i.e. there will be no Doppler effect because relative motion between source and listener is zero.

Case 6: Source and listener moves at right angle to the direction of wave propagation. \[n'=n\]

It means there is no change in frequency of sound heard if there is a small displacement of source and listener at right angle to the direction of wave propagation but for a large displacement the frequency decreases because the distance between source of sound and listener increases.