A plane progressive wave is represented by the
equation: y= 0.1sin(200pi(t)-20pi/15(x)) where
y is the displacement in metres, 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
Comparing the eqn with y=asin2pi(ft)-2piX/L. 2piFt=200pit, makin F the subject gives 200pit/2pit which gives F=100HZ. In finding lamda L, 2pix/L=20pix/15, cross multiply n make L the subject. L= 1.5 speed(v)=L*F, 1.5*100=150m/s
To find the frequency of the wave, we need to determine the coefficient of the "t" term in the equation.
(i) Frequency:
The coefficient of "t" is 200π, which represents the angular frequency of the wave. To find the frequency, we need to divide the angular frequency by 2π.
Frequency = angular frequency / 2π
Frequency = 200π / 2π
Frequency = 100 Hz
(ii) Wavelength:
The coefficient of the "x" term in the equation represents the wave number, which is given as -20π/15. The wave number is related to the wavelength by the formula:
Wave number = 2π / wavelength
Rearranging the equation, we get:
Wavelength = 2π / wave number
Wavelength = 2π / (-20π/15)
Wavelength = -15 / 20
Wavelength = -0.75 m or 0.75 m (We take the absolute value)
(iii) Speed:
The speed of a wave is determined by the equation:
Speed = frequency * wavelength
Speed = 100 Hz * 0.75 m
Speed = 75 m/s
Therefore:
(i) The frequency of the wave is 100 Hz.
(ii) The wavelength of the wave is 0.75 m.
(iii) The speed of the wave is 75 m/s.
To find the frequency, wavelength, and speed of the wave, we can analyze the given equation.
The equation for a plane progressive wave is given by:
y = A sin(kx - ωt),
where A is the amplitude, k is the wave number, ω is the angular frequency, and t is the time. In this case, the equation is given as:
y = 0.1 sin(200πt - (20π/15)x).
Comparing the given equation with the general form, we can deduce the values for the parameters as follows:
(i) Frequency
The frequency of a wave is given by the angular frequency divided by 2π:
Angular frequency (ω) = 200π (from the equation)
Frequency = angular frequency / (2π) = 200π / (2π) = 100 Hz.
Therefore, the frequency of the given wave is 100 Hz.
(ii) Wavelength
The wave number (k) is related to the wavelength by the equation:
k = 2π / λ,
where λ is the wavelength.
By comparing the given equation with the general form, we can find the wave number:
-20π/15 = k,
which can be simplified as:
k = -4π/3.
Now, we can determine the wavelength (λ):
k = 2π / λ,
-4π/3 = 2π / λ.
Solving for λ, we get:
λ = -3 / 2 meters.
Note: The negative sign indicates that the wave is traveling in the negative x-direction.
Therefore, the wavelength of the given wave is 3/2 meters.
(iii) Speed
The speed (v) of a wave is calculated using the equation:
v = λ * f,
where λ is the wavelength and f is the frequency.
From the previous calculations, we have:
λ = 3/2 meters (wavelength)
f = 100 Hz (frequency).
Substituting these values into the equation, we can find the speed:
v = (3/2) * 100 = 150 meters per second.
Therefore, the speed of the given wave is 150 meters per second.
In general, you can model this wave as
y=Amplitude sin 2PI/lambda (x-vt)
or
y=amplitude sin 2PI(x/lambda - ft)
so
y=.1 sin 2PI(10x/15 -100t)
f=100
lambda=15/10=1.5 meters
speed=f*lambda