On a slightly different tack let's go back to the basic derivation of Doppler shift. Suppose the source is moving towards to observer. Then the wavelength is shortened by $V_{source} \cdot \Delta T$ where $\Delta T$ is the reciprocal of sound frequency. Whereby we derive the shift co-efficient:

$$\frac{V_{sound}}{V_{sound}-V_{source}}$$

But what I am struggling to understand is that if rather the observer is moving towards the source, why we may not determine that the wavelength is similarly shortened by $V_{listener} \cdot \Delta T$ and hence derive the shift co-efficient as

$$\frac{V_{sound}}{V_{sound}-V_{listener}}$$

In which case the formula does indeed depend only upon relative motion between source and listener and not whether it is source or observer moving.

On a slightly different tack let's go back to the basic derivation of Doppler shift. Suppose the source is moving towards to observer. Then the wavelength is shortened by $V_\textrm{source} \cdot \Delta T$ where $\Delta T$ is the reciprocal of sound frequency. Whereby we derive the shift co-efficient:

$$\frac{V_\textrm{sound}}{V_\textrm{sound}-V_\textrm{source}}$$

But what I am struggling to understand is that if rather the observer is moving towards the source, why we may not determine that the wavelength is similarly shortened by $V_\textrm{listener} \cdot \Delta T$ and hence derive the shift co-efficient as

$$\frac{V_\textrm{sound}}{V_\textrm{sound}-V_\textrm{listener}}$$

In which case the formula does indeed depend only upon relative motion between source and listener and not whether it is source or observer moving.

On a slightly different tack let's go back to the basic derivation of Doppler shift. Suppose the source is moving towards to observer. Then the wavelength is shortened by V (source) x delta T where delta T is the reciprocal of sound frequency. Whereby we derive the shift co-efficient:

v sound / (v sound - v source).

But what I am struggling to understand is that if rather the observer is moving towards the source, why we may not determine that the wavelength is similarly shortened by V (listener) x delta T and hence derive the shift co-efficient as

v sound / (v sound - v listener).

In which case the formula does indeed depend only upon relative motion between source and listener and not whether it is source or observer moving.

On a slightly different tack let's go back to the basic derivation of Doppler shift. Suppose the source is moving towards to observer. Then the wavelength is shortened by $V_{source} \cdot \Delta T$ where $\Delta T$ is the reciprocal of sound frequency. Whereby we derive the shift co-efficient:

$$\frac{V_{sound}}{V_{sound}-V_{source}}$$

But what I am struggling to understand is that if rather the observer is moving towards the source, why we may not determine that the wavelength is similarly shortened by $V_{listener} \cdot \Delta T$ and hence derive the shift co-efficient as

$$\frac{V_{sound}}{V_{sound}-V_{listener}}$$

In which case the formula does indeed depend only upon relative motion between source and listener and not whether it is source or observer moving.