Stationary waves - why same amplitude etc?

[MBBphys]

Homework Statement

Hi,
In my textbook, it says that waves superposing to form stationary waves, in addition to being in opposite directions, should have the same frequency and ideally the same amplitude - why the 'ideally' and why is having the same amplitude important? Is it because then we don't get nodes of zero displacement? If so, why is that?
Further, it also says that because they have the same frequency, at certain points they are in antiphase; can't two waves of different frequencies be in antiphase at a point though?
Thanks :)

Homework Equations

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The Attempt at a Solution

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[ehild]

Homework Statement

Hi,
In my textbook, it says that waves superposing to form stationary waves, in addition to being in opposite directions, should have the same frequency and ideally the same amplitude - why the 'ideally' and why is having the same amplitude important? Is it because then we don't get nodes of zero displacement? If so, why is that?
Further, it also says that because they have the same frequency, at certain points they are in antiphase; can't two waves of different frequencies be in antiphase at a point though?
Thanks :)

The form of two waves traveling in opposite directions in a medium are y₁=A₁sin(k₁x-ω₁t) and y₂=A₂sin(k₂x+ω₂t). The displacements add up: Y=y₁+y₂. It can happen that Y is zero at a certain place and time, but that point does not stay stationary if the frequencies are different. It is not a standing wave where the nodes are stationary. Assume equal frequencies and wavelengths, Y=A₁sin(kx-ω₁t) +A₂sin(kx+ωt). Apply the addition law for the sines: sin(kx±ωt) = sin(kx)cos(ωt)±cos(kx)sin(ωt). What do you get for the resultant Y?

 

[MBBphys]

The form of two waves traveling in opposite directions in a medium are y₁=A₁sin(k₁x-ω₁t) and y₂=A₂sin(k₂x+ω₂t). The displacements add up: Y=y₁+y₂. It can happen that Y is zero at a certain place and time, but that point does not stay stationary if the frequencies are different. It is not a standing wave where the nodes are stationary. Assume equal frequencies and wavelengths, Y=A₁sin(kx-ω₁t) +A₂sin(kx+ωt). Apply the addition law for the sines: sin(kx±ωt) = sin(kx)cos(ωt)±cos(kx)sin(ωt). What do you get for the resultant Y?

Appreciate your help, but I haven't covered the equations and math you use at AS Level in the UK yet; perhaps a simpler equation? Thanks :)

 

[ehild]

Appreciate your help, but I haven't covered the equations and math you use at AS Level in the UK yet; perhaps a simpler equation? Thanks :)

You need to know basic Trigonometry to understand waves. Hopefully, you will learn it soon. You do not get stationary nodes if the amplitudes are not equal.