Find the Maximum of Superposition of Waves

[TheDemx27]

Homework Statement

Two waves are produced simultaneously on a string of length L = 1 m. One wave has a wavelength λ of 0.5 m. The other wave has a wavelength λ of 0.2 m. The amplitudes of the waves are the same.

At t=0, at what locations x0 is the displacement y(x0) equal to zero? At what locations xm is the displacement y(xm) an extreme (max or min)? How do these locations change with time?

Homework Equations

-

The Attempt at a Solution

The superposition can be written as
y(x,t)=A[sin(4*pi(x-vt))+sin(10*pi(x-vt))]

to find the minimums at t=0:
0=sin(4pi*x)+sin(10pi*x)

this i have solved, but to find the critical points at t=0 would mean I'd have to solve

0=4pi*cos(4pi*x)+10pi*cos(10pi*x)

Which I do not know how to solve algebraically due to the coefficients.

Also I do not know how to describe how the locations would change over time.

 

[kuruman]

The superposition can be written as
y(x,t)=A[sin(4*pi(x-vt))+sin(10*pi(x-vt))]

Not quite. The arguments of the sine functions must be dimensionless quantities.
Also, it would be easier to see what's going on if you used the trig identity for the sum of two signs of equal amplitudes. See here, for example.
http://www.sosmath.com/trig/Trig5/trig5/trig5.html

 

[haruspex]

the sum of two signs of equal amplitudes.

Did you mean the sum of two sines of unequal magnitudes?