Question

Two sinusoidal waves of the same frequency travel in the same direction along a string. If,${\mathbf{y}}_{\mathbf{m}\mathbf{1}}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{cm}\mathbf{,}{\mathbf{y}}_{\mathbf{m}\mathbf{2}}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{cm}\mathbf{.}\mathbf{}{\mathbf{\varphi }}_{\mathbf{1}}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{}\mathbf{and}\mathbf{}\mathbf{}{\mathbf{\varphi }}_{\mathbf{2}}\mathbf{=}\mathbf{\pi }\mathbf{/}\mathbf{2}\mathbf{rad}$what is the amplitude of the resultant wave?

Short answer

The amplitude of the wave is 5.00 cm

Step-by-step solution

The given data

i) Amplitude of first wave,$y_{1m}=3.00\mathrm{cm}\mathrm{or}0.03\mathrm{m}$

ii) Amplitude of second wave,$y_{2m}=4.00\mathrm{cm}\mathrm{or}0.04\mathrm{m}$

iii) The first phase,${\mathrm{\varphi }}_1=0$

iv) The second phase,${\mathrm{\varphi }}_1=\pi /2$

Understanding the concept of phase

A phasor is a vector that rotates around its tail which is pivoted at the origin of a coordinate system. Here${\mathbf{y}}_{\mathbf{1}\mathbf{m}\mathbf{}}\mathbf{and}\mathbf{}{\mathbf{y}}_{\mathbf{2}\mathbf{m}}$represents the amplitude of the two waves and${\mathit{y}}_{\mathbf{m}}$represents the amplitude of the resultant wave.

Calculation of amplitude

The angle made by the phasor of two corresponding waves is $90°$. Using the Pythagorean Theorem we can write,

$$\begin{gathered} y_m^2=y_{1m}^2+y_{2m}^2 \\ =3^2+4^2 \\ =25 \\ {y}_m=5.00\mathrm{cm} \end{gathered}$$

So that the resultant amplitude is $5.00\mathrm{cm}$