Finding the reflected frequency of a wave, such as sound or radar, primarily involves applying the Doppler effect formula, which accounts for the relative motion between the source, the object causing the reflection (the reflector), and the listener or detector.
The reflected frequency is the frequency observed after a wave has bounced off a moving object. This process essentially involves two stages of the Doppler effect:
- From Source to Reflector: The wave emitted by the source is received by the moving reflector at a shifted frequency.
- From Reflector to Listener: The reflector then acts as a new source, re-emitting the wave (at the frequency it just received) back towards the original source or a listener. This re-emitted wave is again shifted in frequency due to the reflector's motion.
The ultimate reflected frequency observed by the listener is the result of this two-step frequency shift.
Understanding the Doppler Effect and Reflection
The Doppler effect describes the change in frequency of a wave for an observer moving relative to its source. When dealing with reflection from a moving object, we use the principles of the Doppler effect, incorporating the speed of the wave and the velocities of the involved parties.
As mentioned in the reference, calculating the frequency perceived by a listener involves variables like:
- The speed of sound (or the wave's speed, denoted as
v). This is the speed at which the wave travels through the medium. - The velocity of the listener (
v_L). How fast and in what direction the observer is moving relative to the medium. - The velocity of the source (
v_S). How fast and in what direction the wave source is moving relative to the medium.
The standard Doppler formula for the frequency f_L heard by a listener from a source emitting frequency f_S is typically given as:
f_L = f_S * (v + v_L) / (v + v_S)
Where:
- Velocities (
v_Landv_S) are positive if the movement is towards each other (reducing the distance) and negative if moving away (increasing the distance).
The reference snippet provides values used in such a calculation:
- Speed of sound: 344
- Velocity of the listener: -20
- Velocity of the source: -30
These values (344 + (-20) divided by 344 + (-30)) illustrate the denominator and numerator terms derived from the speed of sound and the respective velocities in the Doppler formula used to find "the frequency of the listener".
Steps to Calculate Reflected Frequency
For reflection, the Doppler formula is effectively applied twice. Let's outline the typical scenario where the source and listener are the same (e.g., radar gun or sonar on a boat):
- Frequency received by the Reflector (
f_R): The reflector acts as the 'listener' receiving the original frequencyf_S. Its velocity isv_R.
f_R = f_S * (v + v_R) / (v - v_S)(Assuming source moving towards reflector, reflector moving towards source - sign conventions depend on coordinate system) - Frequency received by the Listener (
f_L): The reflector, now oscillating atf_R, acts as a new 'source' moving with velocityv_R. The original source/listener has velocityv_L.
f_L = f_R * (v - v_L) / (v + v_R)(Assuming reflector moving towards listener, listener moving away from reflector - sign conventions depend on coordinate system)
Combining these steps (and simplifying sign conventions for simplicity here, note they depend on the chosen positive direction):
f_L = f_S * [(v + v_R) / (v - v_S)] * [(v - v_L) / (v + v_R)]
In the common case where the source and listener are the same (v_S = v_L):
f_L = f_S * (v + v_R) / (v - v_R) (If reflector is moving towards the source/listener)
Or more generally, considering the relative motion between the source (S), reflector (R), and listener (L), the total Doppler shift leading to the reflected frequency f_L can be derived from the two-step process.
Example Variables from Reference
The values 344, -20, and -30 from the reference illustrate the numerical inputs for v, v_L, and v_S respectively in a Doppler effect calculation leading to "the frequency of the listener". While the snippet doesn't show the full reflection calculation, it highlights the core components (speed of sound, listener velocity, source velocity) essential for applying the Doppler formula in the context of finding frequency shifts, including those resulting from reflection.
| Parameter | Reference Value | Role in Doppler Calculation |
|---|---|---|
Speed of sound (v) |
344 | Wave propagation speed |
Velocity of listener (v_L) |
-20 | Speed/direction of receiver |
Velocity of source (v_S) |
-30 | Speed/direction of emitter |
Using these parameters in the general Doppler formula f_L = f_S * (v + v_L) / (v + v_S) allows for the calculation of the received frequency, which is the core principle behind determining reflected frequency in Doppler systems.
Understanding the sign conventions for velocities (positive towards each other, negative away) is crucial for accurate calculation using the Doppler formula.
