Write the expression for y as a function of x and t in SI units for a sinusoidal wave traveling along a rope in the negative x direction with the following characteristics : A = 8 cm, wavelength = 80cm, f =3hz and y(0,t) = 0 at t = 0.
b) what if? Write an expression for y as a function of x and t for the wave in part (a) assuming y(x,0) =0 at the point x = 10cm
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To find the expression for y as a function of x and t for a sinusoidal wave, we can use the equation:
y(x, t) = A * sin(k * x - ω * t + φ)
Where:
- A is the amplitude of the wave
- k is the wave number (2π divided by the wavelength)
- ω is the angular frequency (2π times the frequency)
- φ is the phase constant
a) Given the characteristics A = 8 cm, wavelength = 80 cm, and f = 3 Hz, we can calculate the values of k, ω, and φ.
First, let's calculate the wave number (k):
k = 2π / λ
k = 2π / 80 cm
Next, let's calculate the angular frequency (ω):
ω = 2π * f
ω = 2π * 3 Hz
Now, let's determine the phase constant (φ):
Since the wave is traveling in the negative x direction, we need to add π to φ in order to shift the wave appropriately.
φ = π
Now we can substitute these values into the equation to get the expression for y as a function of x and t:
y(x, t) = 8 cm * sin((2π / 80 cm) * x - (2π * 3 Hz) * t + π)
This is the desired expression for y as a function of x and t.
b) In part (a), we were given that y(0, t) = 0 at t = 0. Now, we need to find an expression for y as a function of x and t assuming y(x, 0) = 0 at the point x = 10 cm.
To satisfy y(x, 0) = 0 when x = 10 cm, we need to introduce a phase shift to the equation. This phase shift will depend on the value of x.
Since the wave is traveling in the negative x direction, we need to subtract the phase shift from φ to shift the wave appropriately.
Let's calculate the phase shift (Δφ):
Δφ = -2π * (10 cm / 80 cm)
Now, let's substitute this value into the equation:
y(x, t) = 8 cm * sin((2π / 80 cm) * x - (2π * 3 Hz) * t + π - 2π * (10 cm / 80 cm))
Simplifying this equation gives us the expression for y as a function of x and t assuming y(x, 0) = 0 at x = 10 cm:
y(x, t) = 8 cm * sin((2π / 80 cm) * x - (2π * 3 Hz) * t + π - 0.25π)
This is the desired expression for y as a function of x and t, considering the given conditions.