The following two waves are sent in opposite directions on a horizontal string so as to create a standing wave in a vertical plane: y1(x, t) = (8.20 mm) sin(4.00πx - 430πt) y2(x, t) = (8.20 mm) sin(4.00πx + 430πt), with x in meters and t in seconds. An antinode is located at point A. In the time interval that point takes to move from maximum upward displacement to maximum downward displacement, how far does each wave move along the string?

Explanation:

From the information given:

The angular frequency ω = 430 π rad/s

The wavenumber k = 4.00π which can be expressed by the equation:

k = ω/v

∴

4.00 =  430 /v

v = 430/4.00

v = 107.5 m/s

Similarly: k  = ω/v = 2πf/fλ

We can say that:

k = 2π/λ

4.00 π = 2π/λ

wavelength λ = 2π/4.00 π

wavelength λ = 0.5 m

frequency of the wave can now be calculated by using the formula:

f = v/λ

f = 107.5/0.5

f = 215 Hz

Also, the Period(T) = 1/215 secs

The time at which particle proceeds from point A  to its maximum upward displacement  and to its maximum downward displacement  can be computed as t = T/2;

Thus, the distance(x) covered by each wave during this time interval(T/2) will be:

x = v * t

x = v * T/2

x = λ/2

x = 0.5/2

x =  0.25 m