Question

the graphs below describe two waves. the waves are traveling at the same speed. select the graph of the wave with the greater frequency.

Explanation:

Step1: Understand the relationship between frequency and wavelength

The formula for wave speed \(v = f\lambda\), where \(v\) is the speed, \(f\) is the frequency, and \(\lambda\) is the wavelength. Given that \(v\) is the same for both waves (\(v_1=v_2\)), so \(f_1\lambda_1 = f_2\lambda_2\). Frequency \(f\) and wavelength \(\lambda\) are inversely - proportional (\(f=\frac{v}{\lambda}\)). A wave with a smaller wavelength has a greater frequency.

Step2: Compare the wavelengths of the two waves

Count the number of wavelengths in a given position range. For the first graph (top - graph), in the range of \(x = - 10\) to \(x = 10\) (a total length of \(L = 20\) cm), assume the number of full - wavelength cycles \(n_1\). For the second graph (bottom - graph), assume the number of full - wavelength cycles \(n_2\).
Counting the number of crests and troughs:

• In the first graph (top), within the \(x=-10\) to \(x = 10\) range, there are more cycles (more oscillations). If we assume the distance between two consecutive crests (wavelength \(\lambda\)):
• Let's count the number of wavelengths. For the first graph (top), say there are \(n_1 = 5\) wavelengths in the \(x=-10\) to \(x = 10\) range (\(\lambda_1=\frac{20}{5}=4\) cm).
• For the second graph (bottom), say there are \(n_2 = 2\) wavelengths in the \(x=-10\) to \(x = 10\) range (\(\lambda_2=\frac{20}{2} = 10\) cm).

Since \(f=\frac{v}{\lambda}\) and \(v\) is constant, the wave with the smaller wavelength has a higher frequency.

Answer:

The top - graph (the first graph) has a greater frequency.