JEE Main & Advanced Physics Wave Mechanics Some Typical Cases of Doppler's Effect

(1) Moving car towards wall : When a car is moving towards a stationary wall as shown in figure. If the car sounds a horn, wave travels toward the wall and is reflected from the wall. When the reflected wave is heard by the driver, it appears to be of relatively high pitch. If we wish to measure the frequency of reflected sound then the problem.

Can be solved in a different manner by using method of sound images. In this procedure we assume the image of the sound source behind the reflector.

Here we assume that the sound which is reflected by the stationary wall is coming from the image of car which is at the back of it and coming toward it with velocity \[{{v}_{C}}\]. Now the frequency of sound heard by car driver can directly be given as

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

This method of images for solving problems of Doppler effect is very convenient but is used only for velocities of source and observer which are very small compared to the speed of sound and it should not be used frequently when the reflector of sound is moving.

(2) Moving target : Let a sound source S and observer O are at rest (stationary). The frequency of sound emitted by the source is n and velocity of waves is v.

A target is moving towards the source and observer, with a velocity \[{{v}_{T}}\]. Our aim is to find out the frequency observed by the observer, for the waves reaching it after reflection from the moving target. The formula is derived by applying Doppler equations twice, first with the target as observer and then with the target as source.

The frequency \[n'\] of the waves reaching surface of the moving target (treating it as observer) will be \[n'=\left( \frac{v+{{v}_{T}}}{v} \right)\,n\]

Now these waves are reflected by the moving target (which now acts as a source). Therefore the apparent frequency, for the real observer O will be \[n''=\frac{v}{v-{{v}_{T}}}n'\]\[\Rightarrow \]\[n''=\frac{v+{{v}_{T}}}{v-{{v}_{T}}}n\]

(i) If the target is moving away from the observer, then \[n'=\frac{v-{{v}_{T}}}{v+{{v}_{T}}}n\]

(ii) If target velocity is much less than the speed of sound, \[({{v}_{T}}<<v),\] then \[n'=\left( 1+\frac{2{{v}_{T}}}{v} \right)\,n,\] for approaching target

and \[n'=\left( 1-\frac{2{{v}_{T}}}{v} \right)\,n,\] for receding target

(3) Transverse Doppler's effect

(i) If a source is moving in a direction making an angle \[\theta \] w.r.t. the observer

The apparent frequency heard by observer O at rest

At point A : \[{n}'=\frac{nv}{v-{{v}_{S}}\cos \theta }\]

As source moves along AB, value of \[\theta \] increases, \[\cos \theta \] decreases, \[{n}'\]goes on decreasing.

At point C : \[\theta ={{90}^{o}}\], \[\cos \theta =\cos \,{{90}^{o}}=0\], \[{n}'=n\].

At point B : the apparent frequency of sound becomes

\[{n}''=\frac{nv}{v+{{v}_{s}}\cos \theta }\]

(ii) When two cars are moving on perpendicular roads : When car-1 sounds a horn of frequency n, the apparent frequency of sound heard by car-2 can be given as \[n'=n\,\left[ \frac{v+{{v}_{2}}\cos {{\theta }_{2}}}{v-{{v}_{1}}\cos {{\theta }_{1}}} \right]\]

(4) Rotating source/observer : Suppose that a source of sound/observer is rotating in a circle of radius r with angular velocity \[\omega \] (Linear velocity \[{{v}_{S}}=r\omega \])

(i) When source is rotating

(a) Towards the observer heard frequency will be maximum

i.e.  \[{{n}_{\max }}=\frac{nv}{v-{{v}_{S}}}\]

(b) Away from the observer heard frequency will be minimum

and \[{{n}_{\min }}=\frac{nv}{v+{{v}_{S}}}\]

(c) Ratio of maximum and minimum frequency

\[\frac{{{n}_{\max }}}{{{n}_{\min }}}=\frac{v+{{v}_{S}}}{v-{{v}_{S}}}\]

(ii) When observer is rotating

(a) Towards the source heard frequency will be maximum

i.e. \[{{n}_{\max }}=\frac{nv}{v-{{v}_{S}}}\]

(b) Away from the source heard frequency will be minimum

and \[{{n}_{\min }}=\frac{nv}{v+{{v}_{S}}}\]

(c) Ratio of maximum and minimum frequency

\[\frac{{{n}_{\max }}}{{{n}_{\min }}}=\frac{v+{{v}_{S}}}{v-{{v}_{S}}}\]

(iii) Observer is situated at the centre of circle : There will be no change in frequency of sound heard, if the source is situated at the centre of the circle along which listener is moving..

(5) SONAR : Sonar means Sound Navigation and Ranging.

(i) Ultrasonic waves are used to detect the presence of big rocks, submarines etc in the sea.

(ii) The waves emitted by a source are reflected by the target and received back at the SONAR station.

(iii) If v is velocity of sound waves in water and \[{{v}_{sub}}\] is velocity of target (submarine), the apparent frequency of reflected waves will be

\[n'=\left( 1\pm \frac{2{{v}_{sub}}}{v} \right)\,n\]

+ sign is for target approaching the receiver and ? sign for target moving away.