Revision sheet: waves and sound (PHYS 101)

On your formula sheet: this chapter's lines, and when each one holds

• • $v=f\lambda$: The medium sets $v$, the source sets $f$, and $\lambda$ follows. Into another medium, $f$ stays and $\lambda$ changes.
• • $v=\sqrt{F_{T}/\mu}$: $\mu$ is the mass per unit LENGTH in kg/m, not the mass of the string.
• • $\lambda_{m}=2L/m$: A string fixed at both ends, $m=1$, $2$, $3$ and so on, which gives $f_{m}=m\frac{v}{2L}$.
• • $v_{1}\sin\theta_{2}=v_{2}\sin\theta_{1}$: Angles measured from the NORMAL, and the speeds CROSSED: the wave bends toward the normal on the side where it is slower. Divided by $c$, it is $n_{1}\sin\theta_{1}=n_{2}\sin\theta_{2}$.
• • $v_{s}=330$: In m/s, the sheet's typical value for air. When a statement gives another one (343 m/s at $20$ °C), the statement wins.
• • $\beta=10\log_{10}(\frac{I}{I_{0}})=20\log_{10}(\frac{A}{A_{0}})$: Ten with intensities, twenty with amplitudes, because $I$ grows like $A^{2}$: doubling the intensity adds $3$ dB, doubling the amplitude adds $6$ dB.
• • $I_{0}=1.0\times10^{-12}$: In W/m², the threshold of hearing, the reference for $0$ dB.
• • $f_{n}=nf_{1}=n\frac{v}{2L}$: A tube OPEN at both ends: every harmonic, $n=1$, $2$, $3$, $4$ and so on.
• • $f_{n}=nf_{1}=n\frac{v}{4L}$: A tube CLOSED at one end: odd harmonics only, $n=1$, $3$, $5$, $7$ and so on.
• • $f_{b}=|f_{1}-f_{2}|$: Two close frequencies: the ear hears their average pitch, swelling $f_{b}$ times per second.
• • $f'=f\frac{v_{wave}}{v_{wave}\pm v_{source}}$: Moving source: minus when the source approaches (higher pitch), plus when it moves away.
• • $f'=f\frac{v_{wave}\pm v_{obs}}{v_{wave}}$: Moving observer: plus when the observer approaches the source, minus when moving away.

The sheet prints each equation, never the condition under which it holds: copy the condition next to its line on your own sheet before the exam.

The three quantities, and who fixes each one

• • $v = f\lambda$ is the whole chapter, but it is read in one direction only: $f$ comes from the SOURCE, $v$ from the MEDIUM, and $\lambda$ is what is left over.
• • On a string, $v = \sqrt{F/\mu}$ with $F$ in newtons and $\mu$ in kilograms per metre. In air, $v = 331 + 0.60\,T_C$ in metres per second, so $343$ m/s at $20$ degrees Celsius.
• • Snapshot, $y$ against $x$: you read the amplitude and the WAVELENGTH on it. History of one point, $y$ against $t$: you read the amplitude and the PERIOD. A period cannot be measured on a snapshot, which froze time.
• • Crossing into a new medium: $f$ is unchanged, $v$ is the one of the new medium, and $\lambda = v/f$ follows. Each point of the medium only oscillates in place, exactly like the mass on a spring of the previous chapter.

A point sitting at a crest is at the end of its travel and is momentarily at rest; a point crossing $y = 0$ is the one moving fastest. That is the oscillator chapter read again, one oscillator per position.

Refraction: a wave that crosses a boundary at an angle

• • The formula sheet writes it with speeds, under Waves: $v_1\sin\theta_2 = v_2\sin\theta_1$, both angles measured from the NORMAL to the boundary.
• • It comes from the crests alone: they cannot break at the boundary, so they cut it at the same spacing from both sides, $\lambda_1/\sin\theta_1 = \lambda_2/\sin\theta_2$, and the common frequency cancels out of $\lambda = v/f$.
• • Slower on the far side: the wave turns TOWARD the normal. Swell at $12$ m/s reaching shallower water at $7.0$ m/s, $40^{\circ}$ from the normal, leaves at $22.0^{\circ}$, its wavelength going from $120$ m to $70$ m with the same $10$ s period.
• • Checks: $v_2 = v_1$ gives $\theta_2 = \theta_1$, no bending; $\theta_1 = 0$ gives $\theta_2 = 0$, slowed but not turned. The same line, with $v = c/n$, is the refraction of light.

This is why waves break almost parallel to a beach whatever their direction offshore: every depth contour turns them a little more toward the normal.

Transverse and longitudinal waves

• • On a rope each point moves ACROSS the direction of travel: the wave is transverse. In air each layer moves back and forth ALONG it: sound is longitudinal, a train of compressions and rarefactions.
• • The wavelength is read the same way on both: the distance between two neighbouring points in the same state, crest to crest on the rope, compression to compression in air.
• • In both cases what travels at $v$ is the disturbance and its energy; the medium only oscillates about its place.

Standing waves, pipes, levels and shifts

• • String clamped at both ends, or pipe open at both ends: $\lambda_n = 2L/n$ and $f_n = n\,v/(2L)$, every whole $n$. Neighbouring nodes are $\lambda/2 = L/n$ apart.
• • Pipe closed at one end: $\lambda_n = 4L/n$ with $n$ ODD only, so $f_1 = v/(4L)$ and then $3f_1$, $5f_1$, and nothing in between.
• • Two sources, one listening point: count the path difference in wavelengths. $\Delta = n\lambda$ is loud, $\Delta = \left(n+\frac{1}{2}\right)\lambda$ is quiet.
• • Level: $\beta = 10\log_{10}(I/I_0) = 20\log_{10}(A/A_0)$ with $I_0 = 1.0\times 10^{-12}$ W/m$^2$, the second form for an amplitude ratio, and $I = P/(4\pi r^{2})$ for a point source. Beats: $f_{beat} = |f_1 - f_2|$. Doppler: $f' = f\,\dfrac{v \pm v_o}{v \mp v_s}$.

Four numbers to hold instead of four formulas: doubling an intensity adds $3$ dB, doubling an amplitude adds $6$ dB, multiplying an intensity by ten adds $10$ dB, and doubling a distance costs $6$ dB.

Two sources, one listener: the path difference

• • Two sources in step reach a point $P$ by paths $r_1$ and $r_2$. Only the path difference $\Delta = r_2 - r_1$ matters, and only once it is counted in wavelengths.
• • $\Delta/\lambda$ a whole number: crest meets crest, the waves add, loud. A whole number plus a half: crest meets trough, they cancel, quiet.
• • On the line equidistant from the two sources $\Delta = 0$: loud for every frequency at once. Inverting one source swaps every loud point with a quiet one.

Beats: one note that throbs

• • Two close frequencies $f_1$ and $f_2$ sounded together are heard as ONE note at the average $(f_1 + f_2)/2$, whose loudness swells and fades $|f_1 - f_2|$ times a second.
• • Count loudness MAXIMA: the two notes fall back into step $|f_1 - f_2|$ times a second. With $256$ Hz and $260$ Hz that is $4$ swells per second, heard as one note near $258$ Hz, never as a hum at $4$ Hz and never as $2$ swells per second.
• • A beat count alone gives two candidates, $f_{ref} \pm f_{beat}$. Detune on purpose: if tightening a string makes the beats faster, it was already above the reference.
• • Nothing moves in beats, while the Doppler effect needs motion: two steady sources and a slow throb mean beats, one moving source or listener means Doppler.