Mastering Waves: Frequency, Wavelength, and the Wave Equation

What the examiner is testing

The examiner is assessing your ability to calculate an unknown quantity (wave speed, frequency, or wavelength) given the other two, using the wave equation. Marks are awarded for correctly identifying the formula, rearranging it if necessary, substituting values with correct units, and stating the final answer with appropriate units.

The method

1. Identify the known quantities from the question. These will be two of: wave speed (\(v\)), frequency (\(f\)), or wavelength (\(\lambda\)).
2. State the wave equation: \(v = f \times \lambda\).
3. If the unknown quantity is not already isolated, rearrange the equation to solve for it.
4. Substitute the known values, including their units, into the rearranged equation.
5. Calculate the unknown quantity.
6. State your final answer with the correct unit.

Worked example

A sound wave has a frequency of \(250 \text{ Hz}\) and a wavelength of \(1.36 \text{ m}\). Calculate the speed of the sound wave.

$$ \begin{aligned} \text{Knowns: } & f = 250 \text{ Hz} \\ & \lambda = 1.36 \text{ m} \\ \text{Unknown: } & v \\ \text{Equation: } & v = f \times \lambda \\ \text{Substitution: } & v = 250 \text{ Hz} \times 1.36 \text{ m} \\ \text{Calculation: } & v = 340 \text{ m/s} \\ \text{Answer: } & \text{The speed of the sound wave is } 340 \text{ m/s}. \end{aligned} $$
Sanity check: Sound travels at roughly \(340 \text{ m/s}\) in air, so \(340 \text{ m/s}\) is a reasonable answer.

Worked example: a harder one

A radio station broadcasts at a frequency of \(98.2 \text{ MHz}\). Calculate the wavelength of these radio waves. The speed of electromagnetic waves in a vacuum is \(3.0 \times 10^8 \text{ m/s}\).

Initial thought: Use \(v = f \times \lambda\). Rearrange for \(\lambda = v / f\). Substitute \(v = 3.0 \times 10^8 \text{ m/s}\) and \(f = 98.2 \text{ MHz}\).

Why this fails: The frequency is given in megahertz (\(\text{MHz}\)), but the standard unit for frequency in the wave equation is hertz (\(\text{Hz}\)). Directly substituting \(\text{MHz}\) will lead to an incorrect answer.

Corrected approach:
$$ \begin{aligned} \text{Knowns: } & f = 98.2 \text{ MHz} \\ & v = 3.0 \times 10^8 \text{ m/s} \\ \text{Unknown: } & \lambda \\ \text{Convert frequency: } & 1 \text{ MHz} = 1 \times 10^6 \text{ Hz} \\ & f = 98.2 \times 10^6 \text{ Hz} \\ \text{Equation: } & v = f \times \lambda \\ \text{Rearrange: } & \lambda = \frac{v}{f} \\ \text{Substitution: } & \lambda = \frac{3.0 \times 10^8 \text{ m/s}}{98.2 \times 10^6 \text{ Hz}} \\ \text{Calculation: } & \lambda = \frac{3.0 \times 10^8}{9.82 \times 10^7} \text{ m} \\ & \lambda \approx 3.0549898 \text{ m} \\ \text{Answer (to 3 significant figures): } & \text{The wavelength of the radio waves is } 3.05 \text{ m}. \end{aligned} $$
Sanity check: Radio waves are electromagnetic waves, which travel very fast. A wavelength of a few meters is typical for FM radio.

Practice

1. A water wave travels at \(1.5 \text{ m/s}\) and has a wavelength of \(0.5 \text{ m}\). What is its frequency?
2. Light travels through a vacuum at \(3.0 \times 10^8 \text{ m/s}\). If a particular colour of light has a frequency of \(5.0 \times 10^{14} \text{ Hz}\), what is its wavelength?
3. A wave has a frequency of \(20 \text{ Hz}\) and a wavelength of \(15 \text{ cm}\). Calculate its speed.
4. A student measures the speed of ripples on a pond to be \(0.25 \text{ m/s}\). They observe that 10 complete ripples pass a fixed point in \(4.0 \text{ seconds}\). Calculate the wavelength of these ripples.

Answers:

1. \(f = 3.0 \text{ Hz}\)
2. \(\lambda = 6.0 \times 10^{-7} \text{ m}\)
3. \(v = 3.0 \text{ m/s}\)
4. Working for Q4:
   $$ \begin{aligned} \text{Knowns: } & v = 0.25 \text{ m/s} \\ & \text{10 ripples in 4.0 s} \\ \text{Unknown: } & \lambda \\ \text{First, calculate frequency: } & \text{Frequency } (f) = \frac{\text{Number of ripples}}{\text{Time}} \\ & f = \frac{10}{4.0 \text{ s}} = 2.5 \text{ Hz} \\ \text{Equation: } & v = f \times \lambda \\ \text{Rearrange: } & \lambda = \frac{v}{f} \\ \text{Substitution: } & \lambda = \frac{0.25 \text{ m/s}}{2.5 \text{ Hz}} \\ \text{Calculation: } & \lambda = 0.1 \text{ m} \\ \text{Answer: } & \text{The wavelength of the ripples is } 0.1 \text{ m}. \end{aligned} $$

The three mistakes that lose marks

1. Incorrect unit conversion: Forgetting to convert units like \(\text{MHz}\) to \(\text{Hz}\), or \(\text{cm}\) to \(\text{m}\). This leads to answers that are off by powers of ten, e.g., \(305 \text{ m}\) instead of \(3.05 \text{ m}\) for the radio wave example.
2. Incorrect rearrangement of the wave equation: Mixing up the positions of \(v\), \(f\), and \(\lambda\) when solving for an unknown. For example, calculating \(f = v \times \lambda\) when it should be \(f = v / \lambda\). This produces an answer with the wrong magnitude and units.
3. Missing or incorrect units in the final answer: Stating a numerical answer without its unit, or using the wrong unit (e.g., \(\text{m}\) for frequency). This shows a lack of understanding of what the calculated quantity represents.

30-second recap

The wave equation \(v = f \times \lambda\) links wave speed (\(v\), in \(\text{m/s}\)), frequency (\(f\), in \(\text{Hz}\)), and wavelength (\(\lambda\), in \(\text{m}\)). Always ensure units are consistent (e.g., convert \(\text{cm}\) to \(\text{m}\), \(\text{MHz}\) to \(\text{Hz}\)) before calculation. Rearrange the formula carefully to solve for the unknown quantity.