{"id":2451,"date":"2022-03-07T14:31:32","date_gmt":"2022-03-07T05:31:32","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=2451"},"modified":"2023-03-14T16:55:32","modified_gmt":"2023-03-14T07:55:32","slug":"%e3%81%86%e3%81%aa%e3%82%8a","status":"publish","type":"post","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/2451\/","title":{"rendered":"\u3046\u306a\u308a"},"content":{"rendered":"

\u3046\u306a\u308a\u3092\u4e09\u89d2\u95a2\u6570\u306e\u52a0\u6cd5\u5b9a\u7406\u304b\u3089\u8aac\u660e\u3059\u308b\u3068\u3044\u3046\u8a71\u3002<\/p>\n

<\/p>\n

\u3046\u306a\u308a<\/strong><\/span>\u3068\u306f\uff0c\u632f\u52d5\u6570\u304c\uff08\u308f\u305a\u304b\u306b\uff09\u7570\u306a\u308b2\u3064\u306e\u6ce2\u3092\u91cd\u306d\u5408\u308f\u305b\u308b\u3068\u304a\u3053\u308b\u73fe\u8c61\u30022\u3064\u306e\u6ce2\u306e\u632f\u52d5\u6570\u306e\u5dee\u304c\u3046\u306a\u308a\u306e\u632f\u52d5\u6570<\/strong><\/span>\u306b\u306a\u308b\u3002<\/p>\n

\u3053\u308c\u3092\u4e09\u89d2\u95a2\u6570\u306e\u52a0\u6cd5\u5b9a\u7406\u3067\u8aac\u660e\u3059\u308b\u3002\u7c21\u5358\u306e\u305f\u3081\u306b2\u3064\u306e\u30b3\u30b5\u30a4\u30f3\u6ce2\u3092\u8003\u3048\u308b\u3002\u305d\u308c\u305e\u308c\u306e\u632f\u52d5\u6570\u3092 $f_1, \\ f_2$\u00a0 \u3068\u3059\u308b\u3068\uff0c\u30b3\u30b5\u30a4\u30f3\u306e\u52a0\u6cd5\u5b9a\u7406\u304b\u3089\u5f97\u3089\u308c\u308b\u4ee5\u4e0b\u306e\u5f0f<\/p>\n

\\begin{eqnarray}
\n\\cos(\\alpha + \\beta) + \\cos(\\alpha – \\beta) &=& \\cos\\alpha \\cos\\beta\u00a0 {-}\u00a0 \\sin\\alpha \\sin\\beta \\\\
\n&& +\u00a0 \\cos\\alpha \\cos\\beta + \\sin\\alpha \\sin\\beta \\\\
\n&=& 2 \\cos\\alpha \\cos\\beta
\n\\end{eqnarray}<\/p>\n

\u3067\uff0c $\\alpha + \\beta = 2\\pi f_1\u00a0 t, \\ \\ \\alpha – \\beta = 2\\pi f_2\u00a0 t $ \u3068\u304a\u3051\u3070<\/p>\n

\\begin{eqnarray}
\n\\cos (2\\pi f_1 t) + \\cos (2 \\pi f_2 t) &=&
\n2 \\cos \\left(2\\pi \\frac{f_1 + f_2}{2} t\\right)\u00a0 \\cos \\left(2\\pi {\\color{red}\\frac{f_1 – f_2}{2}} t\\right)
\n\\end{eqnarray}<\/p>\n

\u304c\u51fa\u3066\u304f\u308b\u3002\u3053\u308c\u3092\u898b\u308b\u3068\uff0c\u4e00\u77ac\uff0c\u3046\u306a\u308a\u306e\u632f\u52d5\u6570\u306f2\u3064\u306e\u6ce2\u306e\u632f\u52d5\u6570\u306e\u5dee $|f_1 – f_2|$ \u306e\u534a\u5206<\/strong><\/span>\uff1f\u3068\u601d\u3063\u3066\u3057\u307e\u3046\u304c\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u30b0\u30e9\u30d5\u3092\u63cf\u3044\u3066\u307f\u308b\u3068…<\/p>\n

\n
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$\\omega_1 = 2 \\pi f_1 = 2 \\pi \\times 20\\, \\mbox{(Hz)}, \\ \\omega_2 = 2 \\pi f_2 = 2 \\pi \\times 19\\, \\mbox{(Hz)}$ \u3068\u3057\u3066\u30b0\u30e9\u30d5\u3092\u63cf\u3044\u3066\u307f\u308b\u3002<\/p>\n

2\u3064\u306e\u6ce2\u306e\u632f\u52d5\u6570\u306e\u5dee\u306e\u534a\u5206 $\\displaystyle \\frac{|f_1 – f_2|}{2}$ \u306e\u632f\u52d5\u6570\u306e\u6ce2\u3067\u3042\u308b $\\displaystyle 2 \\cos \\left(2 \\pi \\frac{f_1-f_2}{2} t\\right)$ \u3082\u8d64\u7dda\u3067\u63cf\u304f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

\n
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In\u00a0[1]:<\/div>\n
\n
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f1<\/span>:<\/span> 20$\r\nf2<\/span>:<\/span> 19$\r\nom1<\/span>:<\/span> 2*<\/span>%pi<\/span>*<\/span>f1<\/span>$\r\nom2<\/span>:<\/span> 2*<\/span>%pi<\/span>*<\/span>f2<\/span>$\r\n\r\nplot2d<\/span>([<\/span>cos<\/span>(<\/span>om1<\/span>*<\/span>t<\/span>)<\/span> +<\/span> cos<\/span>(<\/span>om2<\/span>*<\/span>t<\/span>)<\/span>, 2*<\/span>cos<\/span>((<\/span>om1<\/span>-<\/span>om2<\/span>)<\/span>\/<\/span>2*<\/span>t<\/span>)]<\/span>, [<\/span>t<\/span>, -<\/span>2<\/span>, 2]<\/span>, \r\n       [<\/span>style<\/span>, lines<\/span>, [<\/span>lines<\/span>, 2]]<\/span>,\r\n       [<\/span>xlabel<\/span>, \"t\"<\/span>]<\/span>, [<\/span>ylabel<\/span>, \"\"<\/span>]<\/span>, \r\n       [<\/span>legend<\/span>, \"\"<\/span>,\"\"<\/span>])<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
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\n\"\"<\/a><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u4e0a\u306e\u30b0\u30e9\u30d5\u3067\u306f\uff0c\u8d64\u7dda\u306e\u5c71\u304b\u3089\u6b21\u306e\u5c71\u307e\u3067\u306e\u6642\u9593\u3067\u3042\u308b\u5468\u671f\u304c $2$ \u79d2\u3067\u3042\u308a\uff0c\u5f93\u3063\u3066\u632f\u52d5\u3059\u308b\u306f $0.5$ Hz\u3002\u3067\u3082<\/p>\n
$\\displaystyle 2 \\cos \\left(2 \\pi \\frac{f_1-f_2}{2} t\\right)$ \u3068\u3044\u3046\u3088\u308a\u3082\uff0c\u3080\u3057\u308d\u305d\u306e\u7d76\u5bfe\u5024 $\\displaystyle 2 \\left| \\cos \\left(2 \\pi \\frac{f_1-f_2}{2} t\\right) \\right|$ \u3060\u3088\u306d\u3047\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[2]:<\/div>\n
\n
\n
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plot2d<\/span>([<\/span>cos<\/span>(<\/span>om1<\/span>*<\/span>t<\/span>)<\/span> +<\/span> cos<\/span>(<\/span>om2<\/span>*<\/span>t<\/span>)<\/span>, 2*<\/span>abs<\/span>(<\/span>cos<\/span>((<\/span>om1<\/span>-<\/span>om2<\/span>)<\/span>\/<\/span>2*<\/span>t<\/span>))]<\/span>, [<\/span>t<\/span>, -<\/span>2<\/span>, 2]<\/span>, \r\n       [<\/span>style<\/span>, lines<\/span>, [<\/span>lines<\/span>, 2]]<\/span>,\r\n       [<\/span>xlabel<\/span>, \"t\"<\/span>]<\/span>, [<\/span>ylabel<\/span>, \"\"<\/span>]<\/span>, \r\n       [<\/span>legend<\/span>, \"\"<\/span>,\"\"<\/span>])<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
<\/div>\n

\n\"\"<\/a><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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\u4e0a\u306e\u30b0\u30e9\u30d5\u3092\u307f\u308b\u3068\uff0c\u8d64\u7dda\u306e\u5c71\u304b\u3089\u6b21\u306e\u5c71\u307e\u3067\u306e\u6642\u9593\u3067\u3042\u308b\u5468\u671f\u306f $1$ \u79d2\uff0c\u3057\u305f\u304c\u3063\u3066\u632f\u52d5\u6570\u306f $1$ Hz\u3002\u78ba\u304b\u306b2\u3064\u306e\u6ce2\u306e\u632f\u52d5\u6570\u306e\u5dee $20 – 19 = 1\\, \\mbox{(Hz)}$ \u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n
\u3068\u3044\u3046\u3053\u3068\u3067\uff0c\u3046\u306a\u308a\u306e\u632f\u52d5\u6570\u306f\uff0c\u4e09\u89d2\u95a2\u6570\u306e\u52a0\u6cd5\u5b9a\u7406\u7684\u898b\u5730\u304b\u3089\u306f\u4e00\u77ac\uff0c2\u3064\u306e\u6ce2\u306e\u632f\u52d5\u6570\u306e\u5dee\u306e\u534a\u5206\u304b\u3082\uff1f\u3068\u601d\u308f\u308c\u305f\u304c\uff0c\u5b9f\u969b\u306e\u6ce2\u5f62\u3092\u307f\u308b\u3068\uff0c\u632f\u5e45\u306e\u5909\u5316\u306f\u305d\u306e\u7d76\u5bfe\u5024\u3092\u3068\u308c\u3070\u3044\u3044\u306e\u3067\uff0c\u7d50\u5c40\u632f\u52d5\u6570\u306f\u305d\u306e2\u500d\u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n
\u3064\u307e\u308a\uff0c\u3046\u306a\u308a\u306e\u632f\u52d5\u6570\u306f\u300c2\u3064\u306e\u6ce2\u306e\u632f\u52d5\u6570\u306e\u5dee\u306e\u534a\u5206\u306e2\u500d\u300d\u3067\u7d50\u5c40\u300c2\u3064\u306e\u6ce2\u306e\u632f\u52d5\u6570\u306e\u5dee\u300d\u306b\u306a\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"
\u3046\u306a\u308a\u3092\u4e09\u89d2\u95a2\u6570\u306e\u52a0\u6cd5\u5b9a\u7406\u304b\u3089\u8aac\u660e\u3059\u308b\u3068\u3044\u3046\u8a71\u3002<\/p>
\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[14,21],"tags":[],"class_list":["post-2451","post","type-post","status-publish","format-standard","hentry","category-maxima","category-21","nodate","item-wrap"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t