Measurements

1. Waves and superposition

Why it matters: sound, light, radio and water waves all obey the same rules, so one set of ideas explains musical instruments, rainbows, cameras and fibre-optic internet. Use the five modes of the lab in the same order as the sections below.

A travelling wave carries a disturbance at speed $v = f\lambda$ (frequency $f$ in hertz times wavelength $\lambda$, the distance between crests). A circular wave from a point source in a ripple tank is $\psi(r,t) = \frac{A}{\sqrt r}\cos(kr - \omega t + \phi)$, with wavenumber $k = 2\pi/\lambda$ and angular frequency $\omega = 2\pi f$ (the $1/\sqrt r$ spreads the energy over a growing circle). Where waves overlap, their displacements add: the principle of superposition.

2. Two-source interference

Two coherent sources (same frequency, with a fixed phase relation) a distance $d$ apart produce, at any point, a path difference $\Delta r = r_2 - r_1$: how much further the wave from one source has travelled. In-phase sources interfere

$$\begin{gathered}\text{constructively when } \Delta r = m\lambda\\ \text{destructively when } \Delta r = \left(m+\tfrac12\right)\lambda\end{gathered}$$

The nodal (destructive) lines are hyperbolae with the sources as foci; far away they approach straight lines at angles $d\sin\theta = (m + \tfrac12)\lambda$. The number of nodal lines on each side is set by $d/\lambda$. A phase difference $\phi$ between the sources shifts the whole pattern, which is the principle behind phased-array antennas and radar.

3. Young's double slit, single slit and gratings

For light the slits act as coherent sources. With slit width $a$, separation $d$ and $N$ slits, the far-field (Fraunhofer) intensity is the product of a single-slit envelope and a multi-slit interference term:

$$\begin{gathered}I(\theta) = I_0\left(\frac{\sin\beta}{\beta}\right)^2\left(\frac{\sin N\alpha}{N\sin\alpha}\right)^2\\ \beta = \frac{\pi a\sin\theta}{\lambda}, \qquad \alpha = \frac{\pi d\sin\theta}{\lambda}\end{gathered}$$
  • Maxima (principal): $d\sin\theta = m\lambda$. For small angles on a screen at distance $L$, the fringe spacing is $\Delta y = \lambda L/d$.
  • Single-slit minima: $a\sin\theta = m\lambda$ ($m \ne 0$). The central maximum has width $2\lambda L/a$, so narrower slits spread light more.
  • Missing orders occur when an interference maximum falls on an envelope zero, i.e. when $d/a$ is an integer ratio.
  • Gratings (large $N$) make the principal maxima very sharp, with $N-2$ weak subsidiary maxima between them. That is why gratings are used in spectrometers: the resolving power is $\lambda/\Delta\lambda = mN$.

The lab plots exactly this formula, so its peaks and zeros match the theory. Switch on photon build-up to see that the pattern is a probability distribution: single photons land at random, but with probability proportional to $I$.

4. Lenses and mirrors

Sign convention used in this lab: "real is positive". Distances are measured from the lens or mirror. The object distance $u$ is positive for a real object; the image distance $v$ is positive for a real image (on the far side of a lens, or in front of a mirror) and negative for a virtual image. The focal length $f$ is positive for converging elements (convex lens, concave mirror) and negative for diverging ones (concave lens, convex mirror).
$$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}, \qquad \text{magnification } |m| = \frac{|v|}{u} = \frac{h_i}{h_o}\ \text{(size ratio)}$$

With real-is-positive, orientation comes from the sign of $v$: a real image ($v>0$) is inverted, a virtual image ($v<0$) is upright. (Many university texts instead use the Cartesian convention, measuring all distances in the direction light travels, which gives $\frac1f = \frac1v - \frac1u$ with $u$ negative for a real object. The physics is identical; only the bookkeeping differs.) For a spherical mirror $f = R/2$.

Principal rays for a thin lens: (1) a ray parallel to the axis leaves through the far focal point $F'$ (or appears to come from the near one for a diverging lens); (2) a ray through the optical centre is undeviated; (3) a ray through the near focal point $F$ leaves parallel to the axis. Where the rays (or their backward extensions) meet is the image.

Two-lens instruments. The image from the first lens is the object for the second. A compound microscope uses a short-focus objective to form a magnified real image just inside the focal length of the eyepiece, which acts as a magnifier: overall $|m| = |m_1||m_2|$. A refracting telescope places the lenses $f_o + f_e$ apart so parallel light leaves parallel (afocal), with angular magnification $f_o/f_e$ (final image inverted).

5. Refraction, Snell's law and total internal reflection

$$n_1\sin\theta_1 = n_2\sin\theta_2, \qquad n = \frac{c}{v}, \qquad \sin\theta_c = \frac{n_2}{n_1}\ (n_1 > n_2)$$

Light bends towards the normal when it slows down. Going from a denser to a less dense medium, beyond the critical angle $\theta_c$ no refracted ray exists and all the light is reflected: total internal reflection. The partial reflection below $\theta_c$ follows the Fresnel equations (the lab shows the unpolarised average); it grows sharply near $\theta_c$.

Optical fibres trap light by total internal reflection at the core–cladding boundary. Rays entering the end face within the acceptance cone are guided:

$$\text{NA} = \sin\theta_{max} = \sqrt{n_{core}^2 - n_{clad}^2}$$

Fibres carry most of the world's internet traffic, including the submarine cables that link the continents.

6. Standing waves and harmonics

Two equal waves travelling in opposite directions add to a standing wave $y = 2A\sin kx\,\cos\omega t$ with fixed nodes (no motion) and antinodes (maximum motion) half a wavelength apart. Fitting whole patterns into a length $L$ gives the allowed frequencies:

$$\begin{gathered}\text{string fixed at both ends (tension } T\text{, mass per metre } \mu\text{):}\\ f_n = \frac{n v}{2L},\quad v = \sqrt{\frac{T}{\mu}},\quad n = 1,2,3,\dots\\ \text{pipe open at both ends: } f_n = \frac{n v}{2L}\\ \text{pipe closed at one end: } f_n = \frac{n v}{4L},\ n = 1,3,5,\dots\end{gathered}$$

The speed of sound in air is about $v \approx 331.3\sqrt{1 + T_C/273.15}$ m/s (343 m/s at 20 °C). A plucked or blown instrument sounds many harmonics at once; their relative strengths give its timbre. Plucking a string at a fraction $p$ of its length gives harmonic amplitudes proportional to $\sin(n\pi p)/n^2$, so harmonics with a node at the plucking point are missing. Open ends are displacement antinodes (pressure nodes); a closed end is a displacement node. Real pipes need a small end correction (about $0.6r$ per open end), which the lab neglects.

Example 1: Young's fringes

Red laser light ($\lambda = 600$ nm) passes through two slits $d = 0.25$ mm apart onto a screen $L = 1.5$ m away.

  1. Fringe spacing $\Delta y = \lambda L/d = (600\times10^{-9})(1.5)/(0.25\times10^{-3}) = 3.6\times10^{-3}$ m $= 3.6$ mm.
  2. Angle of the first maximum: $\sin\theta_1 = \lambda/d = 0.0024$, so $\theta_1 = 0.138^\circ$: small angles really are small.
  3. If each slit is $a = 0.05$ mm wide, the envelope's first zero is at $\lambda L/a = 18$ mm, so about $2 \times 18/3.6 - 1 = 9$ bright fringes fit in the central envelope (the 5th-order fringes coincide with the zeros: missing orders, since $d/a = 5$).

Example 2: a diffraction grating

A grating with 600 lines/mm is lit by a green 532 nm laser.

  1. $d = 1/600$ mm $= 1.667$ µm.
  2. First order: $\sin\theta_1 = \lambda/d = 0.3192$, $\theta_1 = 18.6^\circ$.
  3. Highest order: $m_{max} = \lfloor d/\lambda \rfloor = \lfloor 3.13 \rfloor = 3$, at $\theta_3 = \sin^{-1}(0.958) = 73.3^\circ$.

Example 3: a projector lens

An object is 15 cm from a converging lens with $f = 10$ cm. Using real-is-positive:

  1. $\frac{1}{v} = \frac1f - \frac1u = \frac{1}{10} - \frac{1}{15} = \frac{1}{30}$, so $v = +30$ cm: a real image 30 cm beyond the lens.
  2. $|m| = v/u = 30/15 = 2$: twice the size, and (being real) inverted.
  3. Move the object to 7 cm (inside $f$): $\frac1v = \frac1{10} - \frac17$, $v = -23.3$ cm. Negative means virtual, on the same side as the object, upright and magnified, $|m| = 23.3/7 = 3.3$: a magnifying glass.

Example 4: critical angles and a fibre

  1. Glass to air: $\theta_c = \sin^{-1}(1/1.50) = 41.8^\circ$. Water to air: $48.8^\circ$. Diamond: only $24.4^\circ$, which is why diamonds sparkle: light is trapped and bounces around inside.
  2. Air to water at $40^\circ$: $\sin\theta_2 = \sin40^\circ/1.33$, $\theta_2 = 28.9^\circ$.
  3. A fibre with $n_{core} = 1.48$ and $n_{clad} = 1.46$ has $\text{NA} = \sqrt{1.48^2 - 1.46^2} = 0.242$, an acceptance half-angle of $14.0^\circ$ in air.

Example 5: tuning a guitar string

A high-E string has $L = 0.650$ m and $\mu = 0.40$ g/m. What tension gives E4 = 329.6 Hz?

  1. Required wave speed $v = 2Lf_1 = 2(0.650)(329.6) = 428.5$ m/s.
  2. $T = \mu v^2 = (0.40\times10^{-3})(428.5)^2 = 73.4$ N (the weight of about 7.5 kg).
  3. Second harmonic: $659$ Hz; fretting at half length doubles every frequency (one octave up).

Example 6: open and closed pipes

  1. At 20 °C, $v = 331.3\sqrt{1 + 20/273.15} = 343.2$ m/s.
  2. A 0.50 m pipe open at both ends: $f_1 = v/2L = 343$ Hz, with harmonics 686, 1030 Hz, …
  3. The same pipe closed at one end: $f_1 = v/4L = 172$ Hz, an octave lower, and only odd harmonics (515, 858 Hz, …), which gives the hollow sound of a clarinet-like tone.

Challenges are checked automatically as you work.

Open experiments

  1. Counting nodal lines. In the ripple tank, double $d$ at fixed $\lambda$. How does the number of nodal lines change? Predict it from $d/\lambda$ first.
  2. Colour and spacing. With a double slit, compare violet (400 nm) and red (700 nm). Which spreads more? This is why a grating splits white light into a spectrum with red furthest out.
  3. Real or virtual? Slide the object through $2f$ and $f$ in front of a converging lens and record $v$ and $m$. Plot $1/v$ against $1/u$: a straight line of gradient −1 and intercept $1/f$.
  4. Plucking position. Pluck the string at 0.5 of its length and listen. Which harmonics are missing in the spectrum, and why?