Write an expression for a transverse harmonic wave that has a wavelength of 2.8 m and propagates to the right with a speed of 14.2 m/s. The amplitude of the wave is 0.13 m, and its displacement at t=0 and x=0 is 0.13 m.
Since the displacement at t = 0 equals the amplitude, it is a cosine wave.
f is the frequency: 14.2/2.8 = 5.07 Hz
y = 0.13*cos(2 pi x/(2.8) - 2 pi f t)
= 0.13*cos(2.244 x - 31.86 t)
Well, this wave seems to be in quite a rhythmic mood! Let's start with the basics:
The equation for a transverse harmonic wave can be written as:
y = A * sin(kx - ωt + φ)
Where:
- y is the vertical displacement of the wave at a given point and time,
- A is the amplitude of the wave,
- k is the wave number (2π divided by the wavelength λ),
- x is the horizontal position,
- ω is the angular frequency (2π divided by the period T),
- t is the time, and
- φ is the phase constant.
Given that the wavelength (λ) is 2.8 m and the wave is propagating to the right with a speed of 14.2 m/s, we can find the angular frequency (ω) and wave number (k) using the formulas:
ω = 2π = 2π/λ
The frequency (f) can be calculated by dividing the wave speed by the wavelength:
f = v/λ
Now let's put the values into the equation and see what it looks like:
y = 0.13 * sin((2π/2.8) * x - (2π * v/λ) * t + φ)
But wait, there's more! We don't know the phase constant (φ) yet. Thankfully, we have one last piece of information: the wave's displacement at t=0 and x=0 is 0.13 m. We can use this to determine the phase constant by substituting the known values:
0.13 = 0.13 * sin((2π/2.8) * 0 - (2π * v/λ) * 0 + φ)
Simplifying further:
1 = sin(φ)
Since the wave's displacement is at its maximum (equal to the amplitude) at t=0 and x=0, we can determine that the phase constant (φ) must be 0 degrees or 180 degrees, where the sine function equals 1. Let's take φ = 0 degrees.
Finally, we arrive at the expression for the transverse harmonic wave:
y = 0.13 * sin((2π/2.8) * x - (2π * v/λ) * t)
To express a transverse harmonic wave, we can use the equation:
y = A * sin((2π / λ) * (x - vt + φ))
where:
- y is the displacement of a particle at a given position and time,
- A is the amplitude of the wave,
- λ is the wavelength of the wave,
- x is the position of the particle along the wave,
- v is the speed of the wave, and
- t is the time.
In this case, we are given:
wavelength (λ) = 2.8 m,
speed (v) = 14.2 m/s, and
amplitude (A) = 0.13 m.
Using the given values, the equation becomes:
y = 0.13 * sin((2π / 2.8) * (x - (14.2 * t) + φ))
However, we are also given that the displacement at t=0 and x=0 is 0.13 m. This means that at those points, the wave is at its maximum amplitude. Therefore, we can substitute these values in the equation:
0.13 = 0.13 * sin((2π / 2.8) * (0 - (14.2 * 0) + φ))
Simplifying further, we can cancel out the amplitude:
1 = sin((2π / 2.8) * φ)
From this equation, we can find the phase constant (φ) by finding the inverse sin of both sides:
sin^(-1)(1) = (2π / 2.8) * φ
φ = (2π / 2.8) * sin^(-1)(1)
Finally, substituting this value back into the equation, we get the expression for the transverse harmonic wave:
y = 0.13 * sin((2π / 2.8) * (x - (14.2 * t) + (2π / 2.8) * sin^(-1)(1)))
To write an expression for a transverse harmonic wave, we can use the general equation:
y(x, t) = A * sin(kx - ωt + φ)
Where:
- y(x, t) is the displacement of the wave at position x and time t.
- A is the amplitude of the wave.
- k is the wave number, related to the wavelength λ by k = 2π / λ.
- x is the position along the wave.
- ω is the angular frequency, related to the wave speed v by ω = 2πf = 2π / T = kv, where f is the frequency and T is the period.
- t is the time.
- φ is the phase constant.
Given:
- Wavelength (λ) = 2.8 m
- Wave speed (v) = 14.2 m/s
- Amplitude (A) = 0.13 m
- Displacement at t=0 and x=0 = 0.13 m
To find k, we can use the formula k = 2π / λ:
k = 2π / 2.8
k ≈ 2.24 rad/m
To find ω, we can use the formula ω = kv:
ω = k * 14.2
ω ≈ 31.768 rad/s
Now let's plug in the given values into the equation:
y(x, t) = 0.13 * sin(2.24x - 31.768t + φ)
The displacement of the wave at any point (x, t) can be determined using this expression.