Problem 2.6 (Engel)
Does the superposition y(x,t) = Asin(kx-wt) + 2Asin(kx+wt) generate a standing wave? Simplify the right hand side by
using a trigonometric identity.
Solution
Strategy. A useful identity is: sin(a+/-b)=sin(a)cos(b)+/-cos(a)sin(b)
Execution. y is the sum of two sine functions. Each sine function operates on a sum of terms, so I will expand each
sine function using the identity given above. Then I will add the expansions together to generate y.
Asin(kx-wt) = Asin(kx)cos(wt) - Acos(kx)sin(wt)
2Asin(kx+wt) = 2Asin(kx)cos(wt) + 2Acos(kx)sin(wt)
Adding these expansions gives:
y(x,t) = 3Asin(kx)cos(wt) + Acos(kx)sin(wt)
Notice that while y is a sum of two terms, both of which are a product f(x)g(t), y itself cannot be factored into a spatial
function and a temporal function. This is required if we are going to have spatial nodes and temporal nodes, the
characteristic nodal features of a standing wave. So y is not a standing wave.
Another way to approach this is to find a value of x = xnode that makes y(xnode , t) = 0 for all t, i.e., to find a spatial node.
Let's suppose such a spatial node exists. In that case, for all values of t we have
0 = 3Asin(kxnode )cos(wt) + Acos(kxnode )sin(wt)
0 = 3sin(kxnode )cos(wt) + cos(kxnode )sin(wt)
-cos(kxnode )sin(wt) = 3sin(kxnode )cos(wt)
tan(wt)=-3tan(kxnode )
But this is impossible because the right hand side of the last line is a constant and the left hand side is not. Therefore, a
spatial node does not exist and y is not a standing wave.