← all cardsTopic 11 · Wave Motion · Energy & intensity
Radar / reflection — the d⁴ pattern
● VERBATIM — examined word-for-wordTwo inverse-square trips. (1) I at target = P/4πd². (2) Power intercepted = that × S. (3) Target re-emits fraction k as a NEW point source. (4) Back at transmitter: divide by 4πd² again →
I_return = PkS/(16π²d⁴)What this actually means
Radar is two inverse-square journeys chained together. Outbound: the intensity arriving at the target is P/4πd². Interception: the target's area S turns that intensity into a captured power.
The target then behaves as a brand-new point source, re-emitting a fraction k of what it caught. Return trip: divide by 4πd² again. Multiply the chain out and d⁴ appears in the denominator.
The skill is bookkeeping whose power is whose at each step. Write the four steps as four labelled lines and the 'show that' takes care of itself.
Prove it — watch it be true
- Assemble the four steps of the formula one by one
- Watch the two 4πd² factors multiply into 16π²d⁴
- Check the log-log plot: return signal slope is −4, a one-way trip is only −2