Common nodes of several waves

Nov 2015
4
0
I am interested in the number of common nodes on the x axis of two sinusoidal waves y1 = sin (a1 x + b1) and y2 = sin (a2 x + b2) on a certain distance D from the origin.
Can one write a generalized formula for more than two waves?
 
Apr 2015
1,093
256
Somerset, England
I am interested in the number of common nodes on the x axis of two sinusoidal waves y1 = sin (a1 x + b1) and y2 = sin (a2 x + b2) on a certain distance D from the origin.
Can one write a generalized formula for more than two waves?
Let's start with your formulae.
Neither of these waves have any nodes since they are progressive waves.

So would you like to explain exactly what you really mean?
 
Nov 2015
4
0
What I meant was the number of points in which the two sinusoidal curves intersect on the x axis.

Thank you.
 
Apr 2015
1,093
256
Somerset, England
So you are looking for solutions to the equation

y = sin (a1 x + b1) = sin (a2 x + b2) = 0 ?
 
Nov 2015
4
0
Yes, and if possible a generalization of the type

y = sin (a1 x + b1) = sin (a2 x + b2) = sin (a3 x + b3) = .........= 0

Actually, I am not interested in the values of the solutions, rather in their number on a distance D from origin
 
Last edited:

physicsquest

PHF Helper
Feb 2009
1,426
474
Wouldn't the solution be of the form a x + b = n pi, n = 0,1,2,....
Thus a1 x + b1 = a2 x + b2 if we take n = 0. Since you are looking for "common nodes" as you have explained them, the x value must be the same. Eliminating x will give, b1/a1 = b2/a2.
But you could also have a1 x + b1 = 0, while a2 x + b2 = pi. If you again eliminate x, you would get -(b1/a1) = (pi -b2)/a2. Thus the number of solutions does not look finite.
 
Nov 2015
4
0
Let me reformulate the problem in terms of sinusoidal curves:

A set of sinusoidal curves of wavelength a n - 1 and a n + 1 (where n takes consecutive integer values and a is a constant), originate at different points on the abscissa. Knowing that the curves of wavelength a n - 1 begin at the points (a - 1) n - 1 and the curves of wavelength a n + 1 begin at the points (a - 1) n + 1 , can one write a formula for the number of distinct points at which the above curves intersect the abscissa on a distance d from the origin?