A string has a linear density of 8.1 x 10-3 kg/m and is under a tension of 200 N. The string is 2.9 m long, is fixed at both ends, and is vibrating in the standing wave pattern shown in the drawing. Determine the (a) speed, (b) wavelength, and (c) frequency of the traveling waves that make up the standing wave.

To determine the speed, wavelength, and frequency of the traveling waves that make up the standing wave, we can use the formulas related to wave velocity, wavelength, and frequency.

(a) Speed: The speed of a wave can be calculated using the formula:

v = √(T/μ)

where v is the speed of the wave, T is the tension in the string, and μ is the linear density of the string.

Plugging in the given values:

T = 200 N
μ = 8.1 x 10^-3 kg/m

v = √(200 N / 8.1 x 10^-3 kg/m)

v = √(200 / 8.1 x 10^-3) m/s

Simplifying and calculating the value, we get:

v ≈ 31.24 m/s

Therefore, the speed of the wave is approximately 31.24 m/s.

(b) Wavelength: The wavelength of a standing wave can be determined by measuring the distance between two consecutive nodes (or antinodes). In the given drawing, if the dark dots represent the nodes and the light dots represent the antinodes, we need to measure the distance between any two consecutive nodes (or antinodes).

Let's assume the distance between two consecutive nodes (or antinodes) is L.

So, the wavelength (λ) can be calculated as:

λ = 2L

Given in the question, the length of the string is 2.9 m, and the standing wave pattern shown indicates four nodes and three antinodes. Since there are four nodes and three antinodes, the length between two consecutive nodes (or antinodes) is:

L = 2.9 m / 4

L = 0.725 m

Therefore, the wavelength (λ) is:

λ = 2 * 0.725 m

λ ≈ 1.45 m

Hence, the wavelength of the wave is approximately 1.45 m.

(c) Frequency: The frequency of a standing wave can be determined using the formula:

f = v / λ

where f is the frequency of the wave, v is the speed of the wave, and λ is the wavelength.

Plugging in the values:

v ≈ 31.24 m/s
λ ≈ 1.45 m

f = 31.24 m/s / 1.45 m

Simplifying and calculating the value, we get:

f ≈ 21.55 Hz

Therefore, the frequency of the wave is approximately 21.55 Hz.