Assume a form y1 = A sin(α) for the transverse displacement of the string. Enter an expression for α of a transverse wave on a string traveling along the positive x-direction in terms of its wavenumber k, the position x, its angular frequency ω, and the time t?

α = 2Πft + 2Πx/¶

Explanation:

A transverse wave is one of the type of electromagnetic waves which occurs when the vibration of the medium is perpendicular to the direction of the wave. Mathematically, the wave equation according to the question

y1 = Asin(α) (1)

since its characteristic wave gives a sine wave.

y1 is the vertical displacement

A is the amplitude.

The expression for α of the transverse wave on a string traveling along the positive x-direction in terms of its wavenumber k, the position x, its angular frequency ω, and the time t can be given as;

α = ωt+theta ... (2)

theta is the phase angle. This phase angle is a function of the position x of the wave and the wave number.

theta = kx

where k = 2Πx/¶... (3)

¶ is the wavelength.

ω which is the angular frequency is a function of the frequency (f) of the wave and it is given as;

ω = 2Πf... (4)

Substituting equation 3 and 5 into 2;

α = 2Πft + 2Πx/¶

Therefore,

y1 = Asin(2Πft + 2Πx/¶)

pythagoras' theorem

square root (-5^2+9^2)

square root (25+81)

square root (106)

angle is tan^-1(9/5)

neg x, pos y