Wave Properties

A wave transfers energy from one place to another without transferring matter. The water (the medium) just oscillates about its rest position as the wave passes through it.

Light is an electromagnetic wave and needs no medium, so it crosses the vacuum of space. Sound, water waves and string waves are mechanical waves, they all need a medium of particles to travel through.

In a transverse wave (such as light or a wave on a string) the particles oscillate perpendicular to the wave's direction. In a longitudinal wave (such as sound) they oscillate parallel to it.

Using the wave equation, $v = f\lambda = (500)(0.68) = 340\text{ m/s}$, the everyday speed of sound in air.

Frequency is the reciprocal of the period: $f = \dfrac{1}{T} = \dfrac{1}{0.020} = 50\text{ Hz}$.

The amplitude is the maximum displacement from the rest line, which the wave reaches at $4\text{ cm}$. (The distance from a crest down to a trough is $8\text{ cm}$, which is twice the amplitude, the classic trap.)

The wavelength is the distance for one complete wave, from one crest to the next: $\lambda = 0.40\text{ m}$. (The graph shows two full waves across $0.80\text{ m}$.)

Rearranging the wave equation: $\lambda = \dfrac{v}{f} = \dfrac{340}{170} = 2.0\text{ m}$.

$f = \dfrac{v}{\lambda} = \dfrac{1500}{0.75} = 2000\text{ Hz}$.

$T = \dfrac{1}{f} = \dfrac{1}{250} = 0.0040\text{ s}$.

Sound is a longitudinal wave of compressions and rarefactions. A compression is a region where the particles are bunched close together, giving higher pressure; a rarefaction is where they are spread out, giving lower pressure.

Since $v = f\lambda$ and $v$ is constant, frequency and wavelength are inversely proportional. Doubling $f$ halves $\lambda$.

A wave's energy grows with its amplitude. Its speed, wavelength, frequency and period are all unchanged by the amplitude.

Wave speed depends on the medium, not on the source. When $f$ doubles, the wavelength halves to keep $v = f\lambda$ the same. (Thinking a higher frequency means a faster wave is a common error.)

The period is the time for one complete cycle, from one crest to the next: $T = 0.020\text{ s}$. (The graph shows two full cycles in $0.040\text{ s}$.)

A displacement-distance graph plots displacement against position, so the length of one full wave along the axis is the wavelength. The period and frequency must come from a displacement-time graph, and the speed needs both ($v = \lambda / T$).

Convert the wavelength to metres first: $2.0\text{ cm} = 0.020\text{ m}$. Then $f = \dfrac{v}{\lambda} = \dfrac{340}{0.020} = 17{,}000\text{ Hz}$. (Forgetting to convert cm to m gives the $170\text{ Hz}$ trap.)

The sound travels to the cliff and back, so it covers twice the distance: $2d = vt = (340)(0.40) = 136\text{ m}$, giving $d = 68\text{ m}$. (Forgetting to halve gives the $136\text{ m}$ trap.)

$\lambda = \dfrac{v}{f} = \dfrac{3.0\times10^8}{1.0\times10^8} = 3.0\text{ m}$. Radio waves are electromagnetic, so they travel at the speed of light.

First find the speed: $v = \dfrac{d}{t} = \dfrac{60}{0.30} = 200\text{ m/s}$. Then $\lambda = \dfrac{v}{f} = \dfrac{200}{100} = 2.0\text{ m}$.

The frequency is $f = \dfrac{1}{T} = \dfrac{1}{0.0050} = 200\text{ Hz}$, so $v = f\lambda = (200)(0.40) = 80\text{ m/s}$. (Equivalently $v = \lambda/T$.)

The graph gives the wavelength as $\lambda = 0.40\text{ m}$ (crest to crest). Then $f = \dfrac{v}{\lambda} = \dfrac{12}{0.40} = 30\text{ Hz}$.

The bobbing gives the frequency: $f = \dfrac{15}{30} = 0.50\text{ Hz}$. The crest spacing is the wavelength, $\lambda = 2.0\text{ m}$. So $v = f\lambda = (0.50)(2.0) = 1.0\text{ m/s}$.

Frequency is set by the source and does not change between media. Since $v = f\lambda$, if $v$ increases while $f$ is fixed, the wavelength must increase to match.

The distance graph gives the wavelength ($\lambda = 0.40\text{ m}$) and the time graph gives the period ($T = 0.020\text{ s}$). So $v = \dfrac{\lambda}{T} = \dfrac{0.40}{0.020} = 20\text{ m/s}$. (Reading both lengths off the correct graph is the key skill.)