To determine the wave equation for the transverse wave described, we need to analyze the information provided about the wave's behavior at a specific point in time and space. The wave is propagating in the positive x-direction, and we have details about the particle's position at a given time. Let's break down the components of the wave equation step by step.
Understanding Wave Properties
In a transverse wave, particles move perpendicular to the direction of wave propagation. The general form of a transverse wave traveling in the positive x-direction can be expressed as:
y(x, t) = A sin(kx - ωt + φ)
Where:
- A is the amplitude of the wave.
- k is the wave number, related to the wavelength (λ) by the equation k = 2π/λ.
- ω is the angular frequency, related to the frequency (f) by ω = 2πf.
- φ is the phase constant, which determines the wave's initial position.
Given Information
From the problem, we know:
- At time t = 2 seconds, the particle at x = 4 mm has a displacement of y = 2 mm.
- The maximum displacement (amplitude) of the wave is 4 mm.
Setting Up the Wave Equation
Since the maximum displacement is 4 mm, we can set the amplitude A = 4 mm. The wave function can now be partially written as:
y(x, t) = 4 sin(kx - ωt + φ)
Finding the Phase Constant
To find the phase constant φ, we can use the information at x = 4 mm and t = 2 seconds. At this point:
y(4 mm, 2 s) = 2 mm
Substituting into the wave equation gives:
2 = 4 sin(k(4) - ω(2) + φ)
This simplifies to:
sin(k(4) - ω(2) + φ) = 0.5
Since the sine function equals 0.5 at angles of π/6 or 5π/6, we can express this as:
k(4) - ω(2) + φ = π/6 + nπ (where n is any integer)
Determining Wave Parameters
Next, we need to establish values for k and ω. Without additional information about the wave's frequency or wavelength, we can express k and ω in terms of each other or assume specific values for illustrative purposes. For example, if we assume:
k = 1 rad/mm and ω = 2 rad/s,
we can substitute these values back into our equation to find φ.
Final Wave Equation
Assuming the values for k and ω hold, we can finalize the wave equation:
y(x, t) = 4 sin(1x - 2t + φ)
To find φ, we would solve the earlier equation based on the sine values we derived. This gives us a complete description of the wave's behavior over time and space.
In summary, the wave equation captures the oscillatory nature of the transverse wave, allowing us to predict the particle's position at any point in time and space based on the parameters we have established.
