A wire is made by welding together two metals having different densities. FIGURE P16.59 shows a 2.00-m-long section of wire centered on the junction, but the wire extends much farther in both directions. The wire is placed under 2250 N tension, then a 1500 Hz wave with an amplitude of 3.00 mm is sent down the wire. How many wavelengths (complete cycles) of the wave are in this 2.00-m-long section of the wire?

Wave frequency in both of the sections: $f=1500Hz$

Tension force applied on the wire, $T=2250N$

Linear density for the left part of wire, role="math" localid="1648882920631" ${\mu }_1=9.00g/m=0.009kg/m$

Linear density for the right part of the wire, ${\mu }_2=25g/m=0.025kg/m$

The speed of the wave in the left section of wire,

$$\begin{gathered} v_l=\sqrt{\frac{T}{{\mu }_1}} \\ {v}_l=\sqrt{\frac{2250N}{0.009kg/m}} \\ {v}_l=500m/s \end{gathered}$$

Wavelength for this section,

$$\begin{gathered} {\lambda }_l=\frac{v_l}{f} \\ {\lambda }_l=\frac{500m/s}{1500Hz} \\ {\lambda }_l=0.333m \end{gathered}$$

Thus the number of cycles in the left part,

$$\begin{gathered} n=\frac{1}{{\lambda }_l} \\ n=\frac{1m}{0.333m} \\ n=3cycles \end{gathered}$$

The speed of the wave in the right section of wire,

$$\begin{gathered} v_r=\sqrt{\frac{T}{{\mu }_2}} \\ {v}_r=\sqrt{\frac{2250N}{0.025kg/m}} \\ {v}_r=300m/s \end{gathered}$$

Wavelength for this section,

$$\begin{gathered} {\lambda }_r=\frac{v_r}{f} \\ {\lambda }_r=\frac{300m/s}{1500Hz} \\ {\lambda }_r=0.2m \end{gathered}$$

Number of cycles for this section,

$$\begin{gathered} n=\frac{1}{{\lambda }_r} \\ n=\frac{1m}{0.2m} \\ n=5cycles \end{gathered}$$

For the $2.0m$long section of the wire, a total of $3+5=8$complete cycles are present. So $8$wavelengths of the wave are there.