$v^{\prime}=\left(\frac{v-u_{m}}{v-22}\right) 176 \mathrm{Hz} \quad\left(\because v^{\prime}=\left(\frac{v-v_{0}}{v-v_{s}}\right) v\right)$

$\because v=330 \mathrm{m} / \mathrm{s}$

$\mathrm{v}^{\prime}=\frac{\left(330-\mathrm{u}_{\mathrm{m}}\right)}{308} \times 176 \mathrm{Hz}$         ......$(i)$

The apparent frequency of. siren as heard by the motor cyclist must also be v' because the motorcyclist does not observe any beats. Thus,

$\mathrm{v}^{\prime}=\left(\frac{330+\mathrm{u}_{\mathrm{m}}}{330}\right) \times 165$        ......$(ii)$

$\left(\because v^{\prime}=\frac{v+v_{0}}{v} v\right)$

From eq. $(i),$ and $(ii),$ we get

$\frac{330-\mathrm{u}_{\mathrm{m}}}{308} \times 176=\frac{330+\mathrm{u}_{\mathrm{m}}}{330} \times 165$

Solving it for $u_{m}$, we get

$\mathrm{u}_{\mathrm{m}}=22 \mathrm{m} / \mathrm{s}$