A plane progressive wave is represented by the wave equation:
y=0.1sin(200 ðt-20ðx/17)
Where y is in millimetres, t is in seconds, and x is the distance from a fixed origin, O, in metres. Find:
The frequency of the wave
The wavelength of the wave
The speed of the wave
The phase difference, in radians between a point 0.25 m from ‘O ‘and a point 1.10 m from ‘O’.
The equation of a wave travelling in the same medium with double the amplitude, and double the frequency, but travelling in the opposite direction.
To find the frequency of the wave, we need to identify the coefficient in front of the time variable (t) in the equation. In this case, the coefficient is 200π.
To get the frequency, we can use the formula:
Frequency (f) = angular frequency (ω) / 2π
where angular frequency (ω) is the coefficient in front of the time variable.
In this case, the angular frequency is 200π. Therefore,
Frequency (f) = 200π / 2π = 100 Hz.
So, the frequency of the wave is 100 Hz.
To find the wavelength of the wave, we need to identify the coefficient in front of the position variable (x) in the equation. In this case, the coefficient is 20π/17.
The formula for wavelength (λ) is given by:
Wavelength (λ) = 2π / wave number (k)
where the wave number (k) is the coefficient in front of the position variable.
In this case, the wave number is 20π/17. Therefore,
Wavelength (λ) = 2π / (20π/17) = (2π × 17) / (20π) = 17 / 10 = 1.7 m.
So, the wavelength of the wave is 1.7 m.
To find the speed of the wave, we can use the formula:
Speed (v) = frequency (f) × wavelength (λ).
In this case, the frequency is 100 Hz, and the wavelength is 1.7 m. Therefore,
Speed (v) = 100 Hz × 1.7 m = 170 m/s.
So, the speed of the wave is 170 m/s.
To find the phase difference between two points, we need to find the difference in the arguments of the sine functions at those points.
Given that one point is at 0.25 m from the origin (O) and another point is at 1.10 m from the origin (O), we can calculate the phase difference.
For the point 0.25 m from the origin, the argument of the sine function is:
200πt - 20πx/17 = 200πt - (20π/17) × 0.25 = 200πt - 5π/17.
For the point 1.10 m from the origin, the argument of the sine function is:
200πt - 20πx/17 = 200πt - (20π/17) × 1.10 = 200πt - 22π/17.
The phase difference, in radians, between these points is the difference between these two arguments:
Phase difference = (200πt - 5π/17) - (200πt - 22π/17)
= -5π/17 + 22π/17
= 17π/17
= π radians.
So, the phase difference between a point 0.25 m from 'O' and a point 1.10 m from 'O' is π radians.
To find the equation of a wave traveling in the same medium with double the amplitude, double the frequency, and traveling in the opposite direction, we can modify the given wave equation.
The amplitude of the new wave is double, so the coefficient in front of the sine function becomes 0.2.
The frequency of the new wave is double, so the coefficient in front of the time variable becomes 400π.
The direction of the wave is opposite, so the sign of the position variable changes to negative.
Therefore, the equation of the new wave is:
y = 0.2sin(400πt + 20πx/17).
So, the equation of the wave traveling in the same medium with double the amplitude, double the frequency, and traveling in the opposite direction is y = 0.2sin(400πt + 20πx/17).