Problem set — phasors

Problem set: phasors¶

Exam-style problems. Work them with a pen before touching a computer; the last one is meant to be finished numerically. Solutions and marking rubrics live with the instructor material and are deliberately not on this site.

Throughout, the course convention holds: phasor $\hat{x} = A e^{-\ii\varphi}$ for the oscillation $A\cos(\omega t + \varphi)$.

Problem 1 — the representation, both ways¶

(a) Write the phasors of $3\cos(\omega t)$, $3\sin(\omega t)$, and $3\cos(\omega t - 2\pi/3)$, and draw all three arrows.

(b) A phasor is $\hat{x} = -2\ii$. Write the oscillation $x(t)$ it represents, and state at what time (in units of the period, starting from $t = 0$) the signal first reaches its maximum.

(c) A classmate using the engineering convention writes the phasor of $3\sin(\omega t)$ and gets the complex conjugate of your answer to (a). Explain in one sentence why, and why neither of you is wrong.

Problem 2 — the interference law¶

Two same-frequency oscillations have amplitudes $A_1 = 3$ and $A_2 = 4$ (same units).

(a) What are the largest and smallest amplitudes their sum can have, and at what phase differences do they occur?

(b) Find the phase difference at which the sum has amplitude exactly 5, and notice which famous triangle you have just drawn.

(c) For equal amplitudes $A_1 = A_2 = a$, show that the resultant amplitude is $2a\left|\cos(\delta/2)\right|$, and check both extremes against (a)-style reasoning.

Problem 3 — beats at the tuning fork¶

A violinist tunes the A string against a 440 Hz fork and hears the loudness pulse 2.0 times per second.

(a) What are the two possible frequencies of the string?

(b) The player tightens the string slightly and the pulsing speeds up. Which of the two was it, and what should the player do instead?

(c) The pulsing slows to one beat every 4 s and then the player stops adjusting. Estimate how far from 440 Hz the string is, and how long you would have to listen to notice the residual detuning at all.

Problem 4 — one hundred sources¶

$N = 100$ identical emitters each produce, at a distant detector, a unit-amplitude oscillation of the same frequency.

(a) The emitters are phase-locked ($\varphi_k = 0$ for all $k$). What amplitude and what intensity (in units of one emitter’s) arrive at the detector?

(b) The phases are instead independent and uniformly random. What is the expected intensity? Derive it from the double sum over cross terms, stating exactly which average vanishes.

(c) The random-phase resultant amplitude is itself random: about how large are its typical fluctuations from one realisation to the next, relative to its RMS value? (A qualitative answer with a reason suffices — this is the seed of the speckle phenomenon, module 41.)

Problem 5 — the phasor polygon, numerically¶

Using wavelab.phasors (in the browser laboratory or locally):

(a) Compute the resultant of $N$ unit phasors with phases $\varphi_k = k\,\delta$ for $k = 0, \dots, N-1$ (a fixed phase step — arrows forming a regular polygon) for $N = 6$ and a range of $\delta$. At which values of $\delta$ does the resultant vanish exactly? Compare with $\left|\sin(N\delta/2)/\sin(\delta/2)\right|$.

(b) Repeat with random phases at $N = 10, 100, 1000$, averaging $\left|\text{sum}\right|^2$ over at least 50 seeds, and fit the scaling exponent of RMS amplitude versus $N$.

(c) The pattern in (a) is the diffraction grating of module 31 in disguise; the scaling in (b) is incoherent light. In two sentences, connect each computation to its optics future.