A band playing music at a frequency n is moving towards a wall at a speed v_b . A motorist is following the band with a speed v _m,-. If v is the speed of sound, the expression for the beat frequency heard by the motorist is

The motorist receives two sounds : direct one and that reflected from the wall'  For direct sound,
$$\begin{gathered} {\mathrm{n}}^{\mathrm{\prime }}=\left[\frac{\mathrm{v}+{\mathrm{v}}_{\mathrm{m}}}{\mathrm{v}+{\mathrm{v}}_{\mathrm{b}}}\right]\mathrm{n} \\ \mathrm{For} \mathrm{reflected} \mathrm{sound}, \mathrm{the} \mathrm{reflected} \mathrm{frequency} \mathrm{of} \mathrm{sound} \mathrm{will} \mathrm{be} \\ {\mathrm{n}}^{\prime \prime }=\left(\frac{\mathrm{V}}{\mathrm{V}-{\mathrm{v}}_{\mathrm{b}}}\right)\mathrm{n} \\ \mathrm{This} \mathrm{frequency} \mathrm{is} \mathrm{received} \mathrm{by} \mathrm{the} \mathrm{moving} \mathrm{motorist}. \mathrm{The} \\ \mathrm{apparent} \mathrm{frequency} \mathrm{received} \mathrm{by} \mathrm{him} \mathrm{is} \mathrm{given} \mathrm{by} \\ {\mathrm{n}}^{\prime \prime \mathrm{\prime }}=\left(\frac{\mathrm{v}+{\mathrm{v}}_{\mathrm{m}}}{\mathrm{v}}\right){\mathrm{n}}^{\prime \prime } \\ =\left(\frac{\mathrm{v}+{\mathrm{v}}_{\mathrm{m}}}{\mathrm{v}}\right)\left(\frac{\mathrm{v}}{\mathrm{v}-{\mathrm{v}}_{\mathrm{b}}}\right)\mathrm{n}=\left(\frac{\mathrm{v}+{\mathrm{v}}_{\mathrm{m}}}{\mathrm{v}-{\mathrm{v}}_{\mathrm{b}}}\right)\mathrm{n} \\ \therefore \text{ Beat frequency }={\mathrm{n}}^{"\mathrm{\prime }}-{\mathrm{n}}^{\mathrm{\prime }} \\ =\left(\frac{\mathrm{v}+{\mathrm{v}}_{\mathrm{m}}}{\mathrm{v}-{\mathrm{v}}_{\mathrm{b}}}\right)\mathrm{n}-\left(\frac{\mathrm{v}+{\mathrm{v}}_{\mathrm{m}}}{\mathrm{v}+{\mathrm{v}}_{\mathrm{b}}}\right)\mathrm{n} \\ =\left[\frac{\left(\mathrm{v}+{\mathrm{v}}_{\mathrm{m}}\right)2{\mathrm{v}}_{\mathrm{b}}}{{\mathrm{v}}^2-{\mathrm{v}}_{\mathrm{b}}^2}\right]\mathrm{n} \end{gathered}$$