In the equations of Waves, I find that somewhere they have used the phase constant and somewhere haven't. While deriving the formula of standing wave they assumed two equation as $ y_1\; =\; y_0\, \sin(kx - \omega t)\,$ and $ y_2\; =\; y_0\, \sin(kx +\omega t)\,$ where kx is the phase constant. Then they added it.

But while deriving the equation of beat, they didn't include the phase constant. They assumed that $ y_1\; =\; y_0\, \sin(\omega_1 t)\,$ and $ y_2\; =\; y_0\, \sin(\omega_2 t)\,$ and then added them to get the formula.

It's really confusing me. My understanding says I should always use the phase constant as the displacement depends on it. But clearly the equation of beat is derived without using the phase constant. Why aren't we using it here?


In the case of beat, you're interested in how a pair of waves will 'sound' for a particular observer at some point $x$. So for both waves, $kx$ is a constant and can be ignored.

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