A plane progressive wave is represented by the equation: y= 0.1sin(200pi(t)-20pi(x/17) where y is the displacement in millimetres, t is in seconds and x is the distance from a fixed origin O in metres. Find (i) the frequency of the wave, (ii) its wavelength (iii) its speed, (iv) the phase difference in radians between a point 0.25m from O and a point 1.10m from O.
(i) For the frequency f, require that
200*pi*t = 2*pi*f*t
and solve for f. Obviously f = 100 Hz
(ii) For the wavelength L, require that
20 pi (x/17) = 2 pi (x/L).
That means in this case
10/17 = 1/L
L = 1.7 meters.
(iii) The wave speed is f*L = 170 m/s
(iv) The phase difference can differ by added multiples of 2 pi. I assume you want the minimum phase difference.
The difference between 1.1 and 0.25 is 0.85 m. That is 1/2 of a wavelength, so that would be the phase shift.
pls I need more explanation on this question
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To find the answers to these questions, we need to examine the given plane progressive wave equation. Let's go step by step:
(i) The frequency of the wave can be determined by examining the coefficient of 't' in the sine function. In this case, the coefficient is 200π. Recall that the general equation for a sinusoidal wave is y = A sin(ωt + φ), where ω represents the angular frequency. Since ω = 2πf, where f is the frequency, we can equate ω to 200π and solve for f: 200π = 2πf. By canceling out the common factor of 2π on both sides, we find f = 100 Hz.
(ii) The wavelength of a wave can be obtained by considering the coefficient of 'x' in the sine function. Here, the coefficient is -20π/17. The general formula for the wavelength is λ = 2π/k, where k is the wave number. In this case, k is equal to the coefficient -20π/17: k = -20π/17. Thus, the wavelength λ is calculated as follows: λ = 2π / (-20π/17). Simplifying, we get λ = 17/10 m.
(iii) The speed of a wave can be determined by multiplying the frequency (f) by the wavelength (λ). In this case, we have f = 100 Hz and λ = 17/10 m. The speed (v) is given by the equation v = f * λ: v = 100 * (17/10) = 170 m/s.
(iv) To find the phase difference in radians between two points on the wave, we need to use the distance formula and express it in terms of wavelength. The phase difference (Δφ) is given by Δφ = (2π / λ) * Δx, where Δx is the difference in distance from the fixed origin O. In this case, Δx is 1.10 m - 0.25 m = 0.85 m. Plugging in the values, we get Δφ = (2π / (17/10)) * 0.85 = 10π * 0.85 = 8.5π radians.
To summarize:
(i) The frequency of the wave is 100 Hz.
(ii) The wavelength of the wave is 17/10 meters.
(iii) The speed of the wave is 170 m/s.
(iv) The phase difference in radians between a point 0.25 m from O and a point 1.10 m from O is 8.5π radians.