波、声音与物理光学

A wave 波 carries energy through a medium (or space) without carrying the matter along. A single travelling disturbance is a pulse; a repeating disturbance forms a periodic wave. Two types:

• Transverse 横波: oscillations are perpendicular to the wave's travel. Rope particles move; for light it is the electric and magnetic fields that oscillate.
• Longitudinal 纵波: the medium moves along the direction of travel (sound).

A repeating wave is described by:

• wavelength 波长 $\lambda$ (distance between repeats),
• frequency 频率 $f$ (cycles per second) and period $T=1/f$,
• amplitude 振幅 $A$ (maximum displacement – related to energy),
• wave speed 波速 $v=f\lambda$, set by the medium, not the source.

Worked example. A musical note has frequency $340\ \text{Hz}$ and the speed of sound is $340\ \text{m/s}$. Its wavelength is $\lambda=v/f=340/340=1.0\ \text{m}$. If the same note travels into water (where sound moves at $\approx 1500\ \text{m/s}$) the frequency stays $340\ \text{Hz}$ but the wavelength stretches to $1500/340\approx 4.4\ \text{m}$ – the source sets the frequency, the medium sets the speed and hence the wavelength.

At a boundary a wave partly reflects and partly transmits. Reflecting off a denser medium inverts the wave; off a less-dense medium it does not. Polarization 偏振 applies to transverse waves only: a polarizer passes just one direction of oscillation, which is why polarized sunglasses cut glare.

The effect of a polarizer is measured through intensity 强度 - the average power a wave delivers per unit area over one period, in $\text{W m}^{-2}$. Because a polarizer removes the components of the oscillation that are not aligned with it, it reduces the wave's intensity: unpolarized light passing a single ideal polarizer drops to half its intensity, and a second polarizer at an angle cuts it further. Intensity also falls with distance from a source, since the same power spreads over a larger area.

Electromagnetic waves 电磁波 are oscillating electric and magnetic fields that travel through vacuum at the speed of light $c$, needing no medium. They span the spectrum from radio to gamma rays; higher frequency means shorter wavelength and higher photon energy.

The Doppler effect 多普勒效应 is the change in observed frequency when a wave source and observer move relative to each other. Approaching $\Rightarrow$ higher frequency (shorter wavelength); receding $\Rightarrow$ lower frequency. It explains a passing siren's drop in pitch and the redshift of receding galaxies.

When waves overlap they superpose (add). Constructive interference 相长干涉 (crests aligned) gives a bigger wave; destructive interference 相消干涉 (crest on trough) cancels. Two waves of the same frequency and speed, with equal amplitudes and opposite directions, can form a standing wave 驻波 with fixed nodes 波节 (no motion) and antinodes 波腹 (maximum motion) – the basis of resonance on strings and in pipes.

Worked example. A guitar string $0.65\ \text{m}$ long is fixed at both ends. Its fundamental (first harmonic) fits half a wavelength between the ends, so $\lambda=2L=1.30\ \text{m}$. If waves travel along the string at $260\ \text{m/s}$, the note's frequency is $f=v/\lambda=260/1.30=200\ \text{Hz}$. Shortening the string (a fret) raises the pitch.

When two waves of slightly different frequency overlap, they drift in and out of step, so the combined sound swells loud and soft in a slow throb called beats 拍. The beat frequency 拍频 is simply the difference of the two frequencies:

Two tuning forks of $256\ \text{Hz}$ and $260\ \text{Hz}$ sounded together give $|260-256|=4$ beats each second; a musician tunes an instrument by slowing these beats towards zero against a reference note.

Diffraction 衍射 is the bending and spreading of waves around edges or through openings. The spreading is significant when the opening is comparable to the wavelength – so sound (long wavelength) bends around doorways easily, while light (tiny wavelength) needs a very narrow slit.

Coherent 相干 light through two closely spaced slits produces a pattern of bright and dark fringes. Bright fringes occur where the path difference is a whole number of wavelengths:

A diffraction grating 衍射光栅 has many slits, giving sharp, widely spaced bright lines – useful for separating light into its wavelengths.

Worked example. Light of wavelength $600\ \text{nm}$ passes through two slits $0.20\ \text{mm}$ apart. The first bright fringe ($m=1$) sits at

The tiny angle is why the slits must be very close together and the screen far away to see the fringes clearly.

Light reflecting off the top and bottom of a thin film 薄膜 (soap bubble, oil slick) interferes with itself. Depending on the film's thickness and a possible half-wavelength phase flip on reflection off a denser medium, particular wavelengths interfere constructively – producing the shifting colours you see.

The same physics is put to work in an antireflection coating 增透膜 on camera lenses, glasses, and solar cells: a thin transparent layer whose two reflected waves interfere destructively, so almost no light reflects and more is transmitted. For ideal non-absorbing materials at normal incidence, complete cancellation at one chosen vacuum wavelength $\lambda_0$ requires a film index $n_f=\sqrt{n_{\text{air}}n_{\text{glass}}}$ and thickness $\lambda_0/(4n_f)$, one quarter of a wavelength inside the film. Other wavelengths and incidence angles can still reflect some light.

Answer each before reading the answers below.

1. A wave on a string has frequency $50\ \text{Hz}$ and wavelength $0.80\ \text{m}$. Find its speed.
2. A string $0.90\ \text{m}$ long, fixed at both ends, carries waves at $180\ \text{m/s}$. Find the frequency of its fundamental and of its third harmonic.
3. A $440\ \text{Hz}$ tuning fork and a guitar string sound together and give $3$ beats per second. When the string is tightened slightly, the beats speed up to $5$ per second. What was the string's frequency before it was tightened?
4. Light of wavelength $500\ \text{nm}$ falls on a grating with $600$ lines per millimetre. Find the angles of the first-order and second-order bright lines, and the highest order that can be seen.
5. A student says: "A grating with more lines per millimetre squeezes the bright lines closer together." Correct the student.

1. $v=f\lambda=50\times0.80=40\ \text{m/s}$.
2. $\lambda_1=2L=1.80\ \text{m}$, so $f_1=\dfrac{180}{1.80}=100\ \text{Hz}$. The third harmonic is $3f_1=300\ \text{Hz}$.
3. The string was at $440\pm3=437$ or $443\ \text{Hz}$. Tightening raises its frequency, and the beats got faster, so it moved away from $440\ \text{Hz}$: it was $443\ \text{Hz}$.
4. $d=\dfrac{1}{600}\ \text{mm}=1.67\times10^{-6}\ \text{m}$. First order: $\sin\theta=\dfrac{500\times10^{-9}}{1.67\times10^{-6}}=0.30$, $\theta=17^{\circ}$. Second order: $\sin\theta=0.60$, $\theta=37^{\circ}$. Since $\sin\theta\le1$, $m\le d/\lambda=3.3$, so the third order ($\sin\theta=0.90$, $64^{\circ}$) is the highest.
5. More lines per millimetre means a smaller spacing $d$. From $d\sin\theta=m\lambda$, a smaller $d$ gives a larger $\sin\theta$, so the bright lines spread farther apart (and fewer orders fit).

• Use $v=f\lambda$; the source sets the frequency, and when a wave enters a new medium the frequency stays fixed while the speed and wavelength change.
• Distinguish transverse (vibration perpendicular, can be polarised) from longitudinal (vibration along the travel, e.g. sound).
• Constructive interference needs a path difference of a whole number of wavelengths; destructive needs a half-odd number.
• On a string fixed at both ends the fundamental fits half a wavelength ($\lambda=2L$).
• Diffraction is significant only when the gap is comparable to the wavelength — sound bends round doorways, light needs a very narrow slit.
• Beats: two waves of slightly different frequency throb at $f_{\text{beat}}=|f_1-f_2|$ (e.g. two tuning forks; tuning to zero beats).
• An antireflection coating is a quarter-wavelength-thick layer (index between air and glass) whose two reflections cancel, cutting the reflected light.

Common mistakes

• Saying the frequency of a wave changes when it enters a new medium. The frequency is set by the source; the speed and the wavelength change.
• Taking the string length as one wavelength for the fundamental. A string fixed at both ends has $L = \lambda/2$ in its fundamental mode.
• Using the slit width in the double-slit equation. $d\sin\theta = m\lambda$ uses the spacing between the slits.
• Forgetting the unit prefixes in interference: $1\ \text{nm} = 10^{-9}\ \text{m}$, $1\ \text{mm} = 10^{-3}\ \text{m}$.
• Saying a moving source changes the wave's speed. In the Doppler effect the speed in the medium is unchanged; the observed frequency and wavelength change.
• Saying sound can be polarized. Only transverse waves can be polarized; sound in air is longitudinal.