Superposition: Physics 9702 (Cambridge International AS & A Level)

A stationary (standing) wave is set up on a stretched string of length $1.50\text{ m}$, fixed at both ends. At one instant, the string is observed vibrating with $5$ equally spaced loops along its full length (that is, the string is divided into $5$ equal segments, each one half a wavelength long, with a node at each end and at every point between adjacent loops).

What is the wavelength of this stationary wave?

A student sets up a Young's double-slit experiment using a laser as the light source. Monochromatic light of wavelength $600\text{ nm}$ is incident normally on two narrow slits separated by $0.20\text{ mm}$. The interference pattern is observed on a screen placed $3.00\text{ m}$ from the slits.

(a) State what is meant by two light sources being coherent, and explain why the double slit (illuminated by a single laser) must be used rather than two separate light bulbs if a clear, stable interference pattern is to be observed. [2]

(b) Calculate the fringe spacing (the distance between adjacent bright fringes) observed on the screen. [3]

(c) The student then changes the wavelength of the laser, keeping the slit separation and screen distance the same as before, and measures a new fringe spacing of $12.0\text{ mm}$. Calculate this new wavelength, and state whether or not this light is visible to the human eye. [3]

A violet laser beam of wavelength $450\text{ nm}$ is incident normally on a diffraction grating that has $600$ lines per millimetre. A series of bright maxima is observed on a screen.

(a) Show that the spacing $d$ between adjacent lines (slits) of the grating is $1.67\times10^{-6}\text{ m}$ (to 3 significant figures). [2]

(b) Calculate the angle $\theta$ between the straight-through (zero-order) direction and the first-order ($n=1$) maximum. [3]

(c) Determine the highest order of maximum that can actually be observed with this grating and this wavelength, explaining your reasoning. [3]

Two loudspeakers, $S_1$ and $S_2$, are connected to the same signal generator so that they act as coherent sources, emitting sound of wavelength $0.80\text{ m}$ in phase with each other. At a point $P$, the path from $S_1$ is $2.8\text{ m}$ longer than the path from $S_2$.

What is the phase difference (reduced to a value between $0°$ and $360°$) between the two waves arriving at $P$, and what type of interference occurs there?

A narrow pipe of length $0.75\text{ m}$ is closed at one end and open at the other. A loudspeaker held near the open end emits sound of variable frequency, and the speed of sound in the air inside the pipe is $330\text{ m s}^{-1}$.

(a) State the condition on the air displacement at the closed end, and at the open end, of the pipe that must be satisfied for a stationary wave to form inside it. [2]

(b) Show that the wavelength of the fundamental (lowest-frequency) stationary wave that can form in this pipe is $3.0\text{ m}$, and calculate the corresponding fundamental frequency. [3]

(c) Explain why the second harmonic (at twice the fundamental frequency) cannot be produced in this pipe, and state which harmonic is next produced above the fundamental as the frequency is increased. [2]

(d) Calculate the frequency of this next harmonic identified in (c). [2]

The principle of superposition states that when two or more waves meet at a point, the resultant displacement is the vector (algebraic) sum of the individual displacements of each wave at that point.

At a certain instant, two waves overlap at point $X$. One wave alone would produce a displacement of $+3.5\text{ cm}$ at $X$, and the other wave alone would produce a displacement of $-1.5\text{ cm}$ at $X$.

What is the resultant displacement at $X$ at this instant?

Two dippers, $S_1$ and $S_2$, oscillate in phase in a ripple tank, producing coherent water waves of wavelength $2.4\text{ cm}$. A point $P$ on the water surface is $25.2\text{ cm}$ from $S_1$ and $19.2\text{ cm}$ from $S_2$.

(a) State the condition, in terms of path difference and wavelength $\lambda$, for constructive interference to occur at a point, and the condition for destructive interference to occur. [2]

(b) State two conditions that $S_1$ and $S_2$ must satisfy for a stable, observable interference pattern to be produced on the water surface. [2]

(c) Calculate the path difference between the two waves arriving at $P$. [2]

(d) Determine, showing your working, whether the interference at $P$ is constructive, destructive, or neither. [3]

In a Young's double-slit experiment, light of wavelength $500\text{ nm}$ from two coherent slits separated by $0.40\text{ mm}$ produces bright fringes on a screen $2.00\text{ m}$ away.

What is the distance from the central (zero-order) bright fringe to the third-order bright fringe?

White light, containing all wavelengths from violet ($400\text{ nm}$) to red ($700\text{ nm}$), is incident normally on a diffraction grating that has $300$ lines per millimetre. A spectrum is formed in the first order on each side of the central (zero-order) maximum.

(a) Show that the spacing $d$ between adjacent lines of the grating is $3.33\times10^{-6}\text{ m}$ (to 3 significant figures). [2]

(b) Calculate the angular separation, in the first order, between the violet end ($400\text{ nm}$) and the red end ($700\text{ nm}$) of the spectrum. [4]

(c) State and explain the effect on the angular separation calculated in (b) if a grating with more lines per millimetre is used instead. [2]

A pipe of length $0.85\text{ m}$ is open at both ends. A loudspeaker held near one end emits sound of variable frequency, and the speed of sound in the air inside the pipe is $340\text{ m s}^{-1}$.

(a) State the condition on the air displacement that must be satisfied at each open end of the pipe for a stationary wave to form inside it. [1]

(b) Show that the wavelength of the fundamental (lowest-frequency) stationary wave that can form in this pipe is $1.70\text{ m}$, and calculate the corresponding fundamental frequency. [3]

(c) Calculate the frequency of the third harmonic of this pipe. [2]

(d) State the number of displacement nodes present along the pipe (not counting the ends) when the pipe is vibrating at the third harmonic. [1]