![]() ![]() So two sinusoids at different phases end up producing the effect of a single sinusoid.įor example, here are two sinusoids at the same frequency but with different amplitudes and phases. We can use some standard trigonometric identities to write this asįor some appropriate choice of \(A\) and \(\phi\). More generally, what happens when we play two sinusoids of given amplitudes and phases but the same frequency simultaneously? When we combine the sinusoid \(A_1 \sin(2\pi (ft \phi_1))\) and \(A_2 \sin(2\pi (ft \phi_2))\) to produceĪ_1 \cos(2\pi (ft \phi_1)) A_2 \cos(2\pi (ft \phi_2)) So we see that it is possible for two sinusoids with the same frequency and different amplitudes and at different phases can combine to form a single sinusoid at the same frequency with some new amplitude and phase. Using basic trigonometric identities, the basic sinusoid above can be expressed as a superposition of two different sinusoidsĪ \sin(2\pi (ft \phi)) = A_1 \sin(2\pi ft) A_2\sin(2\pi (ft 1/4)) We can create the sound of a sinusoid with a given amplitude and frequency using a synthesizer and when we have two synthesizers we playing together, the result is the sum of two function formed by summing two functions. We can represent the \(x\)-coordinate of the position at any future time \(t\) by the formula \(\cos(2\pi ft).\) On the other hand, the formula \(\sin(2\pi ft)\) defines the \(y\)-coordinate of the position at a future time \(t\) which is the \(x\)-coordinate phase-shifted by a quarter of a cycle i.e. ![]() moves a distance \(2\pi f\) per second), then in Cartesian coordinates, the position at time \(t\) is given by If our point starts at \((1,0)\) at time \(t=0\) and moves at a speed of \(f\) full cycles of the circle per time unit (i.e. We assume our circle has a radius of 1 unit, making the circumference \(2\pi\). Assuming that the point has moved by an angle \(\theta\) from the point \((1,0)\) on the \(x\)-axis, we call its \(y\)-coordinate the sine of the angle \(\theta\), denoted by \(\sin(\theta)\) and we call its \(x\)-coordinate the cosine of \(\theta\), denoted by \(\cos(\theta).\) ![]() The result is shown in the bottom portion of the figure. At each moment in time, the Cartesian coordinates \((x,y)\) of the point can be recorded, and we can plot either the \(x\) or \(y\) coordinate as a function of time. This quantity is referred to as the sinusoid’s frequency. The speed at which the point rotates about the orign can be measured in terms of the number of complete cycles made per second. ![]()
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