# Common nodes of several waves

#### bob29

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?

#### studiot

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?
Neither of these waves have any nodes since they are progressive waves.

So would you like to explain exactly what you really mean?

#### bob29

What I meant was the number of points in which the two sinusoidal curves intersect on the x axis.

Thank you.

#### studiot

So you are looking for solutions to the equation

y = sin (a1 x + b1) = sin (a2 x + b2) = 0 ?

#### bob29

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

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#### physicsquest

PHF Helper
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.

#### bob29

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?