Question

characteristics of waves
use the information in the graphs to answer the questions.

1. how long is the wavelength of the wave below?
2. how large is the amplitude of the wave below?
3. which wave below has the smaller amplitude, a or b?
4. which wave carries more energy, a or b?
5. what is the same for waves x and y below: amplitude, wavelength, or frequency?
6. which wave has a greater frequency, x or y?
7. which wave has a longer wavelength, x or y?

Explanation:

Step1: Determine wavelength

Wavelength is the distance between two consecutive crests (or troughs). In the first wave graph, from \(x = 0\) to \(x = 4\) (one full cycle), so wavelength \(\lambda=4\) metres.

Step2: Determine amplitude

Amplitude is the maximum displacement from the equilibrium (zero) position. The maximum value in the first wave graph is \(2\) metres, so amplitude \(A = 2\) metres.

Step3: Compare amplitudes of wave A and B

Amplitude of wave A: maximum value is \(2\). Amplitude of wave B: maximum value is \(1\). So wave B has smaller amplitude.

Step4: Relate energy and amplitude

Energy of a wave is proportional to the square of the amplitude (\(E\propto A^{2}\)). Since \(A_{A}=2\) and \(A_{B} = 1\), \(A_{A}>A_{B}\), so wave A carries more energy.

Step5: Analyze waves X and Y

For amplitude: both have maximum value \(2\), so amplitude is same.

Step6: Compare frequencies of X and Y

Frequency \(f=\frac{v}{\lambda}\) (assuming speed \(v\) is constant). Wave X has \(\lambda_{X}=4\) (from \(x = 0\) to \(x = 4\) for one cycle), wave Y has \(\lambda_{Y}=2\) (from \(x = 0\) to \(x = 2\) for one cycle). Since \(f=\frac{1}{\lambda}\) (for same - speed waves), \(f_{Y}>f_{X}\).

Step7: Compare wavelengths of X and Y

As calculated above, \(\lambda_{X}=4\) metres, \(\lambda_{Y}=2\) metres. So wave X has longer wavelength.

Answer:

1. \(4\) metres
2. \(2\) metres
3. Wave B
4. Wave A
5. Amplitude
6. Wave Y
7. Wave X