A-level Physics (Advancing Physics)/Signal Frequencies/Worked Solutions

1. What is the frequency of an X-ray (wavelength 0.5 nm)?

X-rays are electromagnetic waves, so they travel at the speed of light (3 x 10⁸ ms⁻¹).

3 x 10⁸ = f x 0.5 x 10⁻⁹

$f={\frac {3\times 10^{8}}{0.5\times 10^{-9}}}=600\times 10^{15}{\mbox{ Hz}}=600{\mbox{ PHz}}$

2. A sound wave, with a frequency of 44 kHz, has a wavelength of 7.7mm. What is the speed of sound?

V = 44 x 10³ x 7.7 x 10⁻³ = 338.8ms⁻¹

3. What is the fundamental frequency of the following signal?

The big spike on the left is at approximately 750 Hz, so this is the fundamental frequency.

4. Approximately how many harmonics does it contain?

There are 14 other big spikes, plus a few other spikes which may be large enough to be harmonics.

5. The three sine waves sin x°, 4sin(2x-50)° and 0.5sin(3x+120)° are added together to form a signal. What are the frequencies of each of the waves? What is the signal's fundamental frequency? Assume that the waves are travelling at the speed of light, and that 60° = 1mm

sin x° has a wavelength of 360°. Using this, we can calculate the wavelengths of the other two waves, since f(ax) stretches f(x) by the reciprocal of a on the x axis. The frequency of each wave is given by the formula:

$f={\frac {v}{\lambda \ }}={\frac {3\times 10^{8}}{\lambda \ \times 10^{-3}}}$

Wave (y=)	Wavelength (°)	Wavelength (mm)	Frequency (GHz)
sin x°	360	6	50
4sin(2x-50)°	180	3	100
0.5sin(3x+120)°	120	2	150

sin x° has the lowest frequency, so 50 GHz is the fundamental frequency of the signal.