y(x, t) = 0.00001 [cos ([tex]\pi[/tex]x - 400[tex]\pi[/tex]t)]
The general equation of the displacement, y(x,t) of a wave at position x and time t and moving in the positive x-direction is given by;
y(x, t) = A cos ([tex]2\pi kx[/tex] - [tex]2\pi ft[/tex])
or
y(x, t) = A cos (2[tex]\pi[/tex]x/λ - [tex]2\pi ft[/tex]) -------------------(i)
Where;
k is the wave number
f is the frequency and
λ is the wavelength
A is the amplitude of the wave
=> Amplitude, A = 0.010mm
Converting this to meters, we have;
=> A = 0.00001m
=> Frequency, f = 200Hz
=> Wave speed, v = 400m/s
From the wave speed (v), we can get the wavelength (λ) of the wave as follows;
=> v = f x λ
Substitute values of f and v into the equation
=> 400 = 200 x λ
=> λ = 2m
=> Wavelength (λ) = 2m
Substituting the values of λ, A and f into equation (i) gives;
=> y(x, t) = A cos (2[tex]\pi[/tex]x/2 - 2[tex]\pi[/tex](200)t)
=> y(x, t) = 0.00001 [cos ([tex]\pi[/tex]x - 400[tex]\pi[/tex]t)]
Therefore the y-equation for a wave travelling in the positive x-direction with frequency 200 Hz, speed 400 m/s, and amplitude 0.010 mm is given by
=> y(x, t) = 0.00001 [cos ([tex]\pi[/tex]x - 400[tex]\pi[/tex]t)]
where the units of x and t are respectively meters and seconds.
