Vector Formulation

Let

• $\underline{v}_s(t) =$ sound-source velocity as a function of time $t$
• $\underline{v}_l(t) =$ velocity of the listener, if any
• $\underline{x}_s(t) =$ position of source
• $\underline{x}_l(t) =$ position of listener
• $\underline{x}\mathrel{\stackrel{\mathrm{\Delta}}{=}}(x_1,x_2,x_3)^T$ is 3D position

We have velocity related to position by

$$\underline{v}_s= \frac{d}{dt}\underline{x}_s(t) \qquad \underline{v}_l= \frac{d}{dt}\underline{x}_l(t). \protect$$

Consider a Fourier component of the source at frequency $\omega_s$. We wish to know how this frequency is shifted to $\omega_l$ at the listener due to the Doppler effect.