(455) HL Doppler effect equations [IB Physics HL]

Overview of the Doppler Effect

  • The Doppler effect occurs when a source of sound moves relative to an observer.

  • When the source moves towards a stationary observer, wave fronts are compressed, resulting in a smaller wavelength and a higher frequency.

  • Conversely, when the source moves away, the wave fronts are stretched, leading to a larger wavelength and a lower frequency.

Key Equation

  • The fundamental equation is:[ V = F \Lambda ]

    • Where:

      • V = wave speed

      • F = frequency

      • ( \Lambda ) = wavelength

  • Inverse relationship:

    • If wavelength decreases (source approaches), frequency increases.

    • If wavelength increases (source moves away), frequency decreases.

Observed Frequency Equation

  • For a moving source:[ F' = F \times \frac{V}{V \pm U_s} ]

    • Variables:

      • F' = observed frequency

      • F = emitted frequency

      • V = speed of sound (approx. 330 m/s)

      • U_s = speed of the source

    • Adding or subtracting depends on the source's direction:

      • Use the minus sign (V - U_s) when the source approaches (to increase frequency).

Moving Observer Equation

  • For a moving observer:[ F' = F \times \frac{V \pm U_o}{V} ]

    • Where:

      • U_o = speed of the observer

    • Again, the choice between plus or minus depends on desired frequency outcome.

Example Problem

  • A car approaches a stationary observer at 10 m/s; the observed frequency is 450 Hz.

  • Question: What is the emitted frequency F?

  • Given:

    • V = 330 m/s

    • U_s = 10 m/s

  • Substituting into the equation:[ F' = F \times \frac{330}{330 - 10} ]

    • Simplifying: ( F' = F \times \frac{330}{320} \approx 1.03125 F )

    • Solving for F: ( F = \frac{450}{1.03125} \approx 436.364 )

    • Rounded to significant figures: ( F \approx 440 Hz )

Conclusion

  • The Doppler effect equations allow for calculation of observed and emitted frequencies based on the movement of sources and observers.

  • Understanding when to add or subtract (in both equations) is essential for correctly determining the frequency shifts.