Sum-to-Product and Product-to-Sum Formulas: Apply It
Express products as sums.
Express sums as products.
Sound Wave Interference
When two sound waves of different frequencies travel through the same space, they interfere with each other, creating patterns of constructive and destructive interference. These interference patterns can be analyzed using product-to-sum and sum-to-product formulas.
How To: Converting Sum to Product
Identify which sum-to-product formula to use based on the functions involved
Identify [latex]\alpha[/latex] and [latex]\beta[/latex] from your expression
Calculate [latex]\frac{\alpha + \beta}{2}[/latex] and [latex]\frac{\alpha - \beta}{2}[/latex]
Substitute into the formula and simplify
Two tuning forks produce sound waves with frequencies that can be modeled by [latex]\sin(440t)[/latex] and [latex]\sin(446t)[/latex], where [latex]t[/latex] is time in seconds. When both forks sound simultaneously, the combined signal is [latex]\sin(440t) + \sin(446t)[/latex].
Express this sum as a product to analyze the interference pattern.
We have a sum of sines, so we use: [latex]\sin\alpha + \sin\beta = 2\sin\left(\frac{\alpha + \beta}{2}\right)\cos\left(\frac{\alpha - \beta}{2}\right)[/latex] Here, [latex]\alpha = 440t[/latex] and [latex]\beta = 446t[/latex] Calculate the averages: [latex]\begin{aligned} \frac{\alpha + \beta}{2} &= \frac{440t + 446t}{2} = \frac{886t}{2} = 443t \ \frac{\alpha - \beta}{2} &= \frac{440t - 446t}{2} = \frac{-6t}{2} = -3t \end{aligned}[/latex] Substitute into the formula: [latex]\begin{aligned} \sin(440t) + \sin(446t) &= 2\sin(443t)\cos(-3t) \ &= 2\sin(443t)\cos(3t) \end{aligned}[/latex]
Musicians use this beating phenomenon to tune instruments. When two strings are perfectly in tune, the beats disappear!
Two flutes produce tones modeled by [latex]\sin(523t)[/latex] and [latex]\sin(529t)[/latex]. Express the sum as a product and identify the beat frequency.
An audio engineer is working with two modulated signals: [latex]\cos(1200t)[/latex] and [latex]\cos(800t)[/latex]. The product of these signals is [latex]\cos(1200t)\cos(800t)[/latex].
Express this product as a sum to understand the frequency components.
Use the product-to-sum formula for cosines: [latex]\cos\alpha\cos\beta = \frac{1}{2}[\cos(\alpha - \beta) + \cos(\alpha + \beta)][/latex] Here, [latex]\alpha = 1200t[/latex] and [latex]\beta = 800t[/latex] [latex]\begin{aligned} \cos(1200t)\cos(800t) &= \frac{1}{2}[\cos(1200t - 800t) + \cos(1200t + 800t)] \ &= \frac{1}{2}[\cos(400t) + \cos(2000t)] \end{aligned}[/latex]
Express the product [latex]\sin(3000t)\cos(500t)[/latex] as a sum and identify the resulting frequency components.