# Two waves on one string are described by the wave functions y_1=3.0\cos(4.0x-

Two waves on one string are described by the wave functions $$\displaystyle{y}_{{1}}={3.0}{\cos{{\left({4.0}{x}-{1.6}{t}\right)}}}\ {y}_{{2}}={4.0}{\sin{{\left({50}{x}-{2.0}{t}\right)}}}$$ where x and y are in centimeters and t is in seconds. Find the superposition of the waves $$\displaystyle{y}_{{1}}+{y}_{{2}}$$ at the points (a) x = 1.00, t = 1.00; (b) x = 1.00, t = 0.500; and (c) x = 0.500, t = 0 Note: Remember that the arguments of the trigonometric functions are in radians.

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unett
Step 1
We need to find superposition of the waves:
$$\displaystyle{y}_{{1}}={\left({3}\text{cm}\right)}{\cos{{\left({4}{x}-{1.6}{t}\right)}}}$$
$$\displaystyle{y}_{{2}}={\left({4}\text{cm}\right)}{\sin{{\left({5}{x}-{2}{t}\right)}}}$$
Resultant wave is:
$$\displaystyle{y}={y}_{{1}}+{y}_{{2}}$$
Step 2
Calculation:
a) Let us calculate resultant wave at the point x=1cm, t=1s:
$$\displaystyle{y}={y}_{{1}}+{y}_{{2}}$$
$$\displaystyle{y}_{{1}}={\left({3}\text{cm}\right)}{\cos{{\left({4}{x}-{1.6}{t}\right)}}}+{\left({4}{c}{m}\right)}{\sin{{\left({5}{x}-{2}{t}\right)}}}$$
$$\displaystyle={\left({3}\text{cm}\right)}{\cos{{\left({4}\times{1}{c}{m}-{1.6}\times{1}{s}\right)}}}+{\left({4}{c}{m}\right)}{\sin{{\left({5}\times{1}{c}{m}-{2}\times{1}{s}\right)}}}$$
y=-1.65cm
Step 3
b) Let us calculate resultant wave at the point x=1cm, t=0.5s
$$\displaystyle{y}={y}_{{1}}+{y}_{{2}}$$
$$\displaystyle{y}_{{1}}={\left({3}\text{cm}\right)}{\cos{{\left({4}{x}-{1.6}{t}\right)}}}+{\left({4}{c}{m}\right)}{\sin{{\left({5}{x}-{2}{t}\right)}}}$$
$$\displaystyle={\left({3}{c}{m}\right)}{\cos{{\left({4}\times{1}{c}{m}-{1.6}\times{0.5}{s}\right)}}}+{\left({4}{c}{m}\right)}{\sin{{\left({5}\times{1}{c}{m}-{2}\times{0.5}{s}\right)}}}$$
y=-6.02cm
c) Let us calculate resultant wave at the point x=0.5cm, t=0s
$$\displaystyle{y}={y}_{{1}}+{y}_{{2}}$$
$$\displaystyle{y}_{{1}}={\left({3}\text{cm}\right)}{\cos{{\left({4}{x}-{1.6}{t}\right)}}}+{\left({4}{c}{m}\right)}{\sin{{\left({5}{x}-{2}{t}\right)}}}$$
$$\displaystyle={\left({3}{c}{m}\right)}{\cos{{\left({4}\times{0.5}{c}{m}-{1.6}\times{0}{s}\right)}}}+{\left({4}{c}{m}\right)}{\sin{{\left({5}\times{0.5}{c}{m}-{2}\times{0}{s}\right)}}}$$
Result
a) y=-1.65cm;
b) t=-6.02;
c)y=1.15cm;