Final answer:
The resulting wave function from two simultaneously traveling waves in the same medium is not a straightforward sum due to varying frequencies and wavenumbers, and it requires more advanced techniques than are typical for high school physics.
Step-by-step explanation:
When two waves travel simultaneously down a medium and have similar forms, they interfere with each other, and the resulting wave function can often be described as the sum of the two individual wave functions, which is a principle of superposition. Given the individual wave equations of the two waves traveling through a slinky as Ψ₁ (x,t)=0.00460sin(6.00x−354t), and Ψ₂ (x,t)=0.00460sin(7.80x−250t), we must add them to find the resulting wave: Ψᵣ (x, t) = Ψ₁ (x,t) + Ψ₂ (x,t). However, due to the complexity of summing sinusoidal functions with different frequencies and phase constants, we cannot simply add them as algebraic sums unless they have the same frequency and wavenumber (k).
In order to write an expression for the resulting wave when waves have different frequencies or wavenumbers, more advanced techniques involving Fourier analysis may be used, which are beyond high school level physics. Therefore, the resulting wave equation in this specific scenario cannot be easily obtained without additional information or constraints on the nature of the interference between the two waves.