Working With Cstephenmurray Answer Key Standing Waves
Stephen Murray's physics worksheets are one of the few free resources that actually hold up when you try them in a real classroom or study session. The standing waves material is no exception. You grab the worksheet, work through the problems, then check against the key. That sounds straightforward, but there are a few things most people don't catch until they've gone through it twice. The answers live on cstephenmurray.com under his physics section. The site hasn't been redesigned in years, which means the URL structure is stable but the navigation is... minimal. You'll find the standing waves answer key on the same page as the worksheet itself, usually linked near the bottom or posted as a separate PDF. The files are straightforward — no login, no paywall, no newsletter capture. Just click and download. I spent an afternoon last semester trying to track down the right file because the site organizes everything by topic rather than by format. The worksheet and the key are on the same page, but if you're skimming quickly you'll miss the answer section. My workaround was to use browser search (Ctrl+F) for "answer" on the page. Takes about ten seconds and saves you from clicking through three different subpages.
What the Material Actually Covers
The standing waves section deals with harmonic series on strings and in air columns. You'll see problems involving fundamental frequency, overtone numbering, node and antinode placement, and the relationship between wavelength and tube or string length. The math is mostly proportional reasoning — once you know that a closed-open tube has only odd harmonics, or that the nth harmonic on a string satisfies L = n(lambda/2), the calculations are trivial. The trick is setting up the problem correctly. One thing the worksheets do well is force you to draw the wave patterns. I've seen too many students skip the diagram step and then get tripped up on whether a node or antinode sits at a particular end of the medium. Drawing it out every time cuts your error rate significantly. The answer key shows the correct diagrams alongside the numerical work, so use both.
A Problem You Probably Won't See Coming
Several of the worksheet problems involve tubes that are open at both ends versus closed at one end, and the answer key assumes you already know which is which. The wording in some problems is ambiguous — you have to infer the boundary conditions from context. I ran into this with problem 7 in the second set, where the text describes a "tube" without specifying openness. The answer key treats it as closed-open, but if you assume open-open you get a completely different harmonic series and your frequency calculations will be off by a factor of two for every overtone. My fix was to check the diagram. When no diagram is given, I look at the numerical answers in the key and work backward to determine which assumption produces them. In that case, the answer for the third harmonic was 440 Hz with a fundamental of about 147 Hz, which only works for a closed-open tube. It's a small thing but it saved me from marking a student wrong on a technicality when the problem itself was poorly worded.
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Pitfalls to Watch For
Overtone numbering is the biggest source of confusion. In a closed-open tube, the first overtone is the third harmonic, not the second. The worksheets use standard physics terminology, but students trained in music theory or self-studying from other sources often mix this up. The answer key is consistent — it uses the physics convention where "harmonic number" refers to the actual multiple of the fundamental, and "overtone number" refers to the order above the fundamental. Keep that straight. Significant figures matter less than you'd expect here. Murray's answers tend to use two or three significant figures regardless of the input precision. Don't lose points over a rounding difference of 0.1 Hz. More importantly, don't round intermediate values. Keep at least four digits through your calculation and round only at the end. I've seen students get the right method but the wrong final answer because they rounded the wavelength to two figures before multiplying by the harmonic number. The speed of sound assumption. Most problems assume v = 343 m/s unless stated otherwise. A few older versions of the worksheet use 340 m/s. The answer key you're looking at should match the worksheet version. If your numbers are close but not exact, check which speed was used. This comes up most often when comparing answers with a classmate who downloaded from a different mirror or a printed copy from an earlier year.
How to Use the Key Effectively
The biggest mistake I see is people checking their answer after solving only half the problem, or worse, looking at the final number before drawing the diagram. Work the full problem first — diagram, equation setup, calculation, units — then compare. The key shows complete work, not just final answers, so it's useful for checking your method too. If your method matches the key but your answer differs, recheck your calculator entries. Input errors are more common than conceptual mistakes at this level. If your method differs but your answer is the same, you're probably fine. If your method differs and your answer differs, figure out which assumption you made differently — usually boundary conditions or harmonic numbering — and trace it back. The worksheets work best when you do them in order. Problems 1 through 5 establish the basic relationships. Problems 6 onward introduce combined concepts like temperature dependence of wave speed or beat frequency interactions. Skipping ahead means you'll hit walls you wouldn't have if you'd worked through the foundation first.
Limitations of This Resource
Murray's materials are solid for introductory physics, but they don't cover every edge case. There's nothing on standing waves in non-uniform media, no treatment of damping, and minimal coverage of the mathematical derivation from the wave equation. If you need that depth, you'll supplement with a textbook like Halliday and Resnick or HyperPhysics. The worksheets are designed for high school and early college AP Physics levels, so they deliberately avoid the more complex derivations. Another limitation is that the answer key doesn't always explain the reasoning behind each step. It shows the work, but it doesn't justify why you chose a particular equation or how to recognize which scenario you're dealing with. That part requires either class instruction or independent research. I recommend keeping a reference sheet nearby that lists the key formulas: f_n = nv/2L for strings and open-open tubes, f_n = nv/4L for closed-open tubes, and the harmonic relationships for each case. Overall, this is one of the better free standing waves resources available. It's accurate, the problems are well-chosen, and the answer key is complete. Just be aware of the ambiguities in problem wording and the conventions it uses, and you'll get what it offers without much frustration.
